1

A Noise Power Ratio Measurement Method forAccurate Estimation of the Error Vector Magnitude

Karl Freiberger, Student Member, IEEE, Harald Enzinger, Student Member, IEEE,and Christian Vogel, Senior Member, IEEE

Abstract—Error vector magnitude (EVM) and noise power ra-tio (NPR) measurements are well-known approaches to quantifythe inband performance of communication systems and theirrespective components. In contrast to NPR, EVM is an importantdesign specification and is widely adopted by modern com-munication standards such as 802.11 (WLAN). However, EVMrequires full demodulation, whereas NPR excels with simplicityrequiring only power measurements in different frequency bands.Consequently, NPR measurements avoid bias due to insufficientsynchronization and can be readily adapted to different standardsand bandwidths.

We argue that NPR-inspired measurements can replace EVMin many practically relevant cases. We show how to set up thesignal generation and analysis for power ratio-based estimationof EVM in OFDM systems impaired by additive noise, poweramplifier (PA) nonlinearity, phase noise and I/Q imbalance.Our method samples frequency-dependent inband errors viaa single measurement and can either include or exclude theeffect of I/Q mismatch, by using asymmetric or symmetric stop-band locations, respectively. We present measurement resultsusing an 802.11ac PA at different back-offs, corroborating thepracticability and accuracy of our method. Using the samemeasurement chain, the mean absolute deviation from the EVMis less than 0.35 dB.

Index Terms—NPR, EVM, EPR, nonlinearity, phase noise, in-termodulation distortion, power amplifiers, wireless LAN, OFDM

I. INTRODUCTION

ORTHOGONAL frequency division multiplexing(OFDM) is a digital modulation scheme popular in

high-datarate applications like WLAN [1] and LTE [2].These standards define maximum error vector magnitude(EVM) values for the transmitted signal. The EVM capturesthe effect of various transceiver imperfections, e.g., additivenoise, nonlinearity, I/Q mismatch, and phase noise [3]. As aresult, the EVM features a strong correlation with the overall

Author version of the manuscript accepted (December 14, 2016) forpublication in the IEEE Transactions on Microwave Theory and Techniques.Date of current version Jan 12, 2017.

Karl Freiberger (email: freiberger@tugraz.at), Harald Enzinger, and Chris-tian Vogel are with the Signal Processing and Speech Communication Lab(SPSC), Graz University of Technology, 8010 Graz, Austria. Christian Vogelis with the FH JOANNEUM – University of Applied Sciences Graz, 8020Graz, Austria. Significant parts of this work were done when the authors werewith the Telecommunications Research Center Vienna (FTW), 1020 Vienna,Austria. This work was supported by the Austrian Research Promotion AgencyFFG (Project Number 4718971).

c©2017 IEEE. Personal use of this material is permitted. Permission fromIEEE must be obtained for all other uses, in any current or future media,including reprinting/republishing this material for advertising or promotionalpurposes, creating new collective works, for resale or redistribution to serversor lists, or reuse of any copyrighted component of this work in other works.

system performance in terms of the bit error rate (BER) [4].The correlation of EVM and BER can also be understoodas a consequence of two principles of EVM measurements:First, EVM uses the actual communication signal as a testsignal, exciting the device under test (DUT) in the sameway as during regular operation, which is crucial in case ofa nonlinear DUT [5]. Second, EVM compares demodulateddata symbols based on the Euclidean distance of the observeddata symbols from the ideal symbols. Consequently, the EVMis a measure of inband error that allows to use symbol-basedcorrection algorithms. In typical receivers the effect oflinear filtering and common phase error is minimized byapplying equalization and de-rotation, respectively. These twocorrection steps are also mandatory for EVM measurementsaccording to the 802.11 standard [1].

Although the principle of EVM makes it a sought-after andpowerful system level metric, it also comes with the followingdrawbacks. First, EVM is sensitive to synchronization errors,e.g., frequency and timing offsets and mismatches [6]. Second,an EVM test setup is expensive [4]. Receiver functionalityis required to test a transmitter, and vice versa. Testing aseparate DUT, e.g., an RF PA, even requires a full transceiver.Advancing to a new standard or bandwidth typically requiresa costly update of the EVM analyzer. Third, it is requiredthat the whole measurement chain operates linearly at thefull signal bandwidth. This gets increasingly difficult withincreasing bandwidth [7], limiting the measurement floor atwide bandwidths.

The noise power ratio (NPR) [8]–[11] is a method that doesnot need demodulated symbols to quantify the inband error. Asa result, the NPR does not suffer from the EVM’s drawbacksoutlined above. Consequently, NPR is an appealing alternativeto EVM in many practical scenarios, e.g., as an optimization-objective for digitally enhanced RF systems [12] such asdigital predistorters [13]. However, although the NPR is awell established metric on its own, literature does not providesufficient support for the following hypothesis: The principlebehind NPR allows for accurately estimating EVM in OFDMsystems in case of typical RF transmitter impairments likephase noise, IQ imbalance, and power amplifier nonlinearity.

Rather, there is prior research indicating that the NPR isnot able to correctly estimate the inband error caused by anonlinear DUT [9], [14], discussed in more detail in Sec. II,below. The only work known to us that explicitly addressesthe relation of EVM and NPR [15] does not cover the issues,we will show to be relevant for EVM estimation, e.g., thebandwidth and location of the stop bands of the NPR test

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signal. The relation of EVM and NPR for IQ imbalance andphase noise has not been addressed at all so far.

We introduce the term error power ratio (EPR) for ourNPR-inspired EVM estimation method, because we define theEPR as an error to signal ratio, just like the EVM, inverselyto the typical NPR definition. Furthermore, we link somespecifics to the term EPR, e.g., required properties of the testsignal to obtain accurate EVM estimates for OFDM signals,which further discerns the EPR from the traditional NPR.

In summary, to the best of our knowledge, this paper is thefirst to present:• The effect of IQ mismatch and phase noise on the NPR.• An analysis and guidance how to setup NPR (EPR)

measurements to accurately estimate EVM in OFDM sys-tems with phase noise, IQ mismatch and power amplifiernonlinearity.

• The usage of multiple stop-bands for resolving smoothfrequency-dependent errors with a single measurement.

• A straightforward procedure to obtain an NPR test signalpreserving the statistics of the OFDM signal.

• Measurement results comparing NPR (EPR) with EVM,over a wide range of inband error levels, using a com-mercial hardware WLAN EVM analyzer as reference.

The remainder of this paper is organized as follows. SectionII discusses related research. In Section III, we define theEPR and propose respective test signal generation and analysismethods. In Section IV, we present typical transmitter impair-ments and their influence on the EPR and EVM along withsimulation results supporting our findings. Our experimentaltest setup and the analysis of hardware measurement resultsis presented in Section V. We discuss uncertainty and bias inSection VII. Section VII concludes this paper. In Appendix A,we define and discuss OFDM and EVM.

II. FOUNDATIONS OF NPR MEASUREMENTS ANDRELATED RESEARCH

In the following, we discuss prior work on NPR measure-ments and EVM estimation. We emphasize why our hypothesisis relevant and that it is neither sufficiently supported, norcontradicted in prior work.

Classic NPR measurements use band-limited Gaussian noisewith a flat power spectral density (PSD) within the band ofinterest and apply a notch (or band-reject, or stop-band) filterto obtain the test signal [9], [10]. A typical DUT fills thestop-band with error, e.g., intermodulation distortion due to anonlinear DUT [16]. The NPR is obtained by comparing theDUT output power without the filter, or outside the notch, tothe power within the notch, i.e., the NPR is defined as a signal(plus error) to error ratio. If the only source of error is additivewhite gaussian noise (AWGN), NPR and EVM are inverselyrelated via the signal to noise ratio (SNR). If, however, theDUT includes nonlinearity, phase noise or modulator imper-fections, the inband error depends on the test signal excitingthe DUT. Since the NPR test signal must include at least oneinband notch, it cannot be identical to the EVM test signal.Therefore, it is not obvious that measurements of NPR can beused to replace EVM, in cases when the inband error depends

on the test signal. Rather, there is prior research indicatingquite the opposite [9], [14].

Pedro et al. [14] argue that the NPR underestimates the in-band error up to 7 dB, because they view correlated co-channeldistortion as a relevant source of error. However, Geens et al.[10] reach the conclusion that the approach in [14] is only validif the input to the nonlinear system varies a lot in amplitude.In most practical cases however, this is not the case and NPRis a good figure of merit for the inband error [10]. In [17],Pedro et al. note that for systems where an equalizer gets ridof dynamic linear effects, and hence the correlated co-channeldistortion, the NPR is a good choice to assess the inband error.Gharaibeh [9] shows that the NPR overestimates the inbanderror of a (quasi-) memoryless nonlinearity by up to 10 dBif the excitation signal differs significantly from a circularly-symmetric complex normal (CSCN) distribution, where thein-phase (I) and quadrature (Q) component are independentand identically distributed (i.i.d) Gaussian [18]. However, theNPR is shown to be a good estimate of inband distortion, ifthe signal approaches a CSCN distribution. In OFDM systems,equalizers are ubiquitous and the signal is CSCN distributed.Consequently, [9], [10], [14] do not contradict our hypothesis.

Traditional NPR measurements use an analog noise sourceand an analog filter to generate the test signal [19]. Morerecently, digital test signal generation has been proposed [11],allowing for better repeatability and lower test times [8]. Amultitone signal is generated via inverse fast Fourier transform(IFFT) of a large number (1k-10k) of tones with randomphases. Equal magnitudes are used in [11], apart from “5 to10% of the tones” in the center of the bandwidth that areset to zero to form the stop band. As outlined above, it isdesirable to have a test signal matching the statistics of thecommunication signal of interest. Therefore, the higher ordercross PSDs of a multitone can be optimized [5], which ishowever computationally demanding [20]. For matching thedistribution of each individual sample of an OFDM signal,such an optimization is not necessary, because summing alarge number of random phase tones yields independent CSCNdistributed samples. However, OFDM uses a cyclic prefix andoversampling may be applied via zero-padding before the IFFT[21]. Consequently, the samples are not independent. Rather,there is correlation, manifesting in a PSD different to the PSDobtained from a signal with random phases as discussed above.

The shape of the PSD is important when assessing the out-of band behavior, e.g., spectral mask [1] or adjacent channelleakage ratio (ACLR) [2]. It is desirable to have an NPRsignal matching the out-of-band PSD of the OFDM signal,because this allows for measuring the inband error (NPR) andthe out-of-band performance with the same test signal. Withestablished NPR test signals [11] this is not advisable, becausethey are strictly bandlimited.

EVM estimation has also been addressed in the context ofpass/fail tests with automatic test equipment (ATE) [22], [23]with the goal to reduce measurement time and cost. Thesemethods differ from our approach by the fact that the type ofDUT is assumed to be known, and an approved prototype ofthe DUT is available. The prototype is used to train surrogatemodels and decision (pass/fail) rules.

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SourceFilter

Multi-Stop DUTAnalysis

EPRs

Source Signal

x

EPR Test Signal

y

DUT Output

1

10 10 10

PSD

(dB

)

Frequency (MHz)Frequency (MHz)Frequency (MHz)

EPR

Source signal Ss(f) EPR test signal Sx(f) DUT output Sy(f)

M3, see Fig. 2−90−90−90

−45−45−45

−20−20−20

0

0

0

0

0

0 101010 202020

Fig. 1. Principle of the error power ratio (EPR). The EPR is our approach to estimate EVM based on the principle of NPR. The source s is the communicationsignal of interest. Depicted is a WLAN signal with 20 MHz bandwidth. We obtain the test signal x by digitally filtering s to attenuate parts of the inband,achieving steep stop-band notches. The device under test (DUT) introduces errors, which rise the power in the stopbands of the DUT output y. The EPR isthe ratio between average power in the stopbands and average power in the inband of y.

1

ωa,3 ωa,3 ωc,3 ωb,3 ωb,3

M3 M3P3 P4

Subcarrier Index

BWstop,3

BWstop,3Sy(ω)

Sx(ω)

PSD

(dB

)

6 6.5 7 7.5 8

−80

−60

−40

−20

0

Fig. 2. Close-up of the third stopband M3 from Fig. 1. The baseband center frequency is at 2.1875 MHz, i.e., ωc,3 = 7∆f2π, with the WLAN OFDMsubcarrier spacing being ∆f = 312.5 kHz. We choose the stopband-width for integrating the error power as BWstop = ∆f . The filter stopband M3 withbandwidth BWstop = 1.3∆f is chosen to be slightly wider to account for the practically limited filter slope steepness.

III. ERROR POWER RATIO

As outlined in the introduction, we use the term error powerratio (EPR) to distinguish our EVM estimation method fromthe traditional NPR [8], [16]. The principle of the EPR isdepicted in Fig. 1. In Fig. 2, we zoom into the third stopbandand illustrate the notation used in the definition of the EPRbelow.

A. Notation

Consider a baseband signal x(t) modulated to an angu-lar carrier frequency ω0 = 2πf0, where t denotes time.The corresponding radio frequency signal is xRF(t) =R{x(t) exp ω0t}, where 2 = −1. The real and imaginarypart of x(t) are denoted by R{x(t)} = xI(t), and I{x(t)} =xQ(t), respectively. In cases where it is not relevant whetherwe are in baseband or RF, we simply use the notation x(t).For discrete time signals, we use the notation x[n]. We useboldface notation for vectors e.g., ω = [ω1, . . . , ωN ]T , whereT denotes transposition.

The power spectral density (PSD) of an ergodic, widesense stationary random signal x(t) is given as Sx(ω) =∫∞−∞Rx(τ)e−ωτdτ , with the auto correlation function de-

fined as the expectation Rx(τ) = E {x∗(t)x(t+ τ)} . Wedenote the average PSD of x(t) over a frequency-set K as

Sx(K) =1

λ(K)

∫ω∈K

Sx(ω) dω , (1)

where λ(K) is the Lebesque measure of the set K, i.e., theoverall bandwidth of the integration.

B. Definition

We define the EPR of a signal x(t) as

EPR{x(t)} =Sx(M)

Sx(P), (2)

i.e., we compare the PSD of x(t) averaged over differentfrequency sets M and P . Here, M is the set of stop-bandfrequencies

M =

NM⋃m=1

Mm , (3)

i.e., the union of NM ∈ N : NM ≥ 1 individual stop-bandsMm. We define the m-th stop-bandMm as the set of angularfrequencies satisfying

Mm = {ω ∈ R : ωa,m ≤ ω < ωb,m} , (4)

where ωa,m = ωc,m− BWstop,m

2 and ωb,m = ωc,m +BWstop,m

2are the start and stop frequency of the m-th stop-band withcenter frequency ωc,m and stop-band bandwidth BWstop,m,respectively, as illustrated in Fig. 2. Analogously to (4), the setof present bands is denoted as P =

⋃NPp=1 Pp, with Pp = {ω ∈

R : ωa,p ≤ ω < ωb,p}. To compare the EPR of individual

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stopbands with subcarrier-dependent EVM values as definedin (38) in the Appendix A-B, we define

EPR(km) =Sx(Mm)

Sx(P), (5)

where Mm has a center frequency at ωc,m = km∆f , wherekm is the OFDM subcarrier index belonging to the mthstopband center, and ∆f is the subcarrier spacing. With equalstopband bandwidth BWstop,m = ∆f , the overall EPR asin (2) is the mean over the subcarrier-dependent EPRs, i.e.,

1NM

∑NMm=1 EPR(km).

Our goal is to use the EPR of the DUT response y(t), i.e.,EPR{y(t)} for estimating the EVM. For that purpose, twoconstraints are necessary in the definition of (2):

1) Sufficient test-signal stop-band rejection: The EPR ofthe test signal x(t) used to excite the DUT must be smallerthan the expected error vector power (EVP) to estimate. TheEVP is the square of the EVM and defined in AppendixA-B. As depicted in Fig. 2 our test signal generation andanalysis achieves approximately −90 dB stopband rejection,which is well below the expected EVM of typical transceivercomponents. A stopband rejection of RM = −10 log10 (ε) dBmeans

Sx(M) = εSx(P) . (6)

2) Reasonable selection of P and M: To achieve require-ment 1), i.e., small EPR{x(t)}, and to provide maximumaveraging over frequency, P should cover the whole signalbandwidth with exclusion of the stop-band and transition-band frequencies M, as indicated in Fig. 2. As discussedin Sec. IV, a good choice for the individual stop-bandwidthsis the OFDM subcarrier-spacing, i.e., BWstop,m = ∆f . Theoptimum number and center frequencies of the stop-bandsMm is problem-dependent. Setting Nm too high, i.e. usingto many missing bands increases the likelihood of altering thesignal statistics significantly. By setting Nm = 1 frequency-dependent errors cannot be resolved in a single measurementand there is only little averaging of missing band power,which increases the variance of the estimator. A convenientchoice for Nm and ωc is using the pilot-tone locations ofthe OFDM signal standard, e.g., subcarriers [−21,−7, 7, 21]for the 20–MHz WLAN signal in Fig. 1. This way, smoothfrequency-dependent errors can be resolved and an increase inbandwidth increases the number of stop-bands. Furthermore,the EPR can be readily compared with the EVM averaged onlyover the pilots which is provided by typical EVM analyzers.However, one has to be aware that by using the symmetricpilot locations as stop-band centers, IQ mismatch is excludedfrom the EPR, as shown in Sec. IV. We further discuss theplacement of the stopbands in Sec. IV and Sec. VI-B.

C. Signal Generation

We use a digital approach to generate a test signal x[n].In a measurement scenario where an analog signal x(t) isrequired, we copy the signal vector vX = {x[n]} to thememory of a signal generator which performs the requireddigital to analog conversion (DAC). Similar to Reveyrandet al. , [11], [16], we generate vX by means of an IFFT,

i.e., vX = ifft(vXf). However, instead of using constantamplitudes and random phases as in [11], we obtain thefrequency-domain vector vXf via an FFT of the communi-cation signal vector vX, and set those bins that correspondto stopband frequencies ω ∈ M to zero before the IFFT,i.e., vXf = fft(vS); vXf(vIndexNull) = 0. Here,M ⊃M indicates that we set broader bands to zero than theactual integration range M used to compute the EPR in (2).M includes the same center frequencies ωc,m asM in (4). Theonly difference is that we use BWstop,m = Kstop BWstop,m,Kstop ∈ R : Kstop ≥ 1 to form M, as shown in Fig. 2.If not stated otherwise, we use Kstop = 1.3. This way, weaccommodate for the limited steepness of the PSD analysiswindow, see Sec. III-D, below. Furthermore, using broaderstop-bands increases the robustness against clock frequencyoffset.

The proposed multiplication in the frequency domain withzero at the bins to be nulled, and one otherwise, correspondsto a circular (cyclic) convolution in the time domain. Thisis welcome in measurement applications, because a typicalsignal generator repeats the signal vector continuously, i.e., itis periodically extended. In contrast to repeating a linearly-filtered signal vector, no discontinuities occur due to therepetition of the circularly-filtered signal vector. Consequently,we do not require synchronization when analyzing the signal.

In Sec. II, we have already highlighted the importance ofusing a test signal matching the statistics of the communicationsignal and that OFDM signals are CSCN distributed. A testsignal xNPR[n] generated with the method in [16] approachesa CSCN distribution, because the phases are independentrandom variables. Since a CSCN signal is invariant to lin-ear filtering [18], also our method delivers a CSCN signal.We denote the test signal generated with our method withxEPR[n]. However, even if the distributions of the individualsamples of two signals are the same, the PSDs can be quitedifferent. A signal xNPR[n] generated with i.i.d random phasesdoes not include possible correlations due to cyclic prefix,windowed overlap and oversampling via IFFT zero padding.Consequently, the PSD Sx,NPR(f) is strictly band-limited, asshown in Fig. 3. In contrast, the PSD of our EPR test signalSx,EPR(f) resembles the PSD Ss(f) of the communicationsignal s(t). Consequently, out-of-band measurements usingan EPR test signal generated with our approach lead toresults well comparable with those made with the actualcommunication signal s(t). Summing up, our signal generationapproach allows for measurements of out-of-band metrics, e.g.,spectral mask [1] or ACLR [2], with the same signal that isused for measuring the inband error.

D. Signal Analysis

To obtain the EPR from the DUT output signal, we proposethe following two approaches. The first is based on PSD-estimation via analyzing the baseband time-domain (TD) DUToutput, whereas the second uses a swept-tuned (ST) spectrumanalyzer to measure the DUT output power in different bands.The advantage of the TD method is that the same analyzercode can be used in measurements and simulations. The ST

5

1

Frequency (MHz)

dB

Sx,NPR(f)

Ss(f)Sx,EPR(f)

Mask

−20

−40

−60

−80

−100

−5

0

0 5 10 15 20

Fig. 3. PSDs of three different test signals generated and analyzed on a PC(not measured). Ss(f) is the PSD estimate of a 320-symbol WLAN signalwith 20 MHz bandwidth oversampled in the IFFT to 60 MHz. In contrastto the traditional NPR signal generation resulting in a strictly band-limitedSx,NPR(f), our EPR test signal Sx,EPR[n] features the same out-of-bandbehavior as the original communication signal Ss(f).

method unites all the advantages outlined in Sec. I, whenmaking measurements. In particular, the ST method does notrequire analog to digital converters (ADCs) able to sample theentire signal bandwidth with high accuracy.

1) Time Domain (TD) Approach: Given

y =[y[n], y[n− 1], . . . , y[n−N + 1]

]T, (7)

i.e., N samples of the digitized baseband-equivalent of theDUT output y[n] = y(nTs), where fs = 1

Tsis the sample rate

in Hz, we use Welch’s periodogram method [24] to estimatethe PSD Sy(ωk) on a discrete frequency grid ωk = k2πfs

K ,K ∈ N, k ∈ Z : d−K/2e ≤ k ≤ dK/2e−1, where d·e denotesthe ceiling operator. The number of averaged periodograms isNavg = bN−KKolap

c+ 1, where K is the FFT length, Kolap is thewindow overlap in samples, and b·c is the floor operator. Oncewe have Sy(ωk) the EPR is computed like in (2), however, dueto discrete frequency grid the integrals turn into sums.

In our WLAN application, we typically use a resolutionof fs/K = 10 kHz, a four-term minimum sidelobe Nuttallwindow [25], and 50% overlap, i.e., Kolap = d0.5Ke. This isalso the setting we use in all our PSD plots, e.g., in Fig. 3.Note that if the resolution was too high, e.g., 50 kHz insteadof 10 kHz, we would not be able to resolve the steep stopbandnotches anymore. For the same reason, a window with goodside-lobe behavior is crucial for the analysis. Given the PSDestimate, we just need to implement (2) by averaging over therespective PSD frequency bins to obtain the EPR.

As can be seen from Fig. 3, our PSD analysis setupcan resolve EPRs down to −90 dB, which is enough fortypical transceiver building blocks. With a rectangular analysiswindow-length exactly matching the IFFT-length of the signalgeneration, there is theoretically no lower limit on the EPR,i.e., the bins set to zero remain zero if there is no error. Sincethe power is integrated in (2), the lack of averaging is not aproblem in practice. However, a slight mismatch of synthesisand analysis window-length, e.g., due to mismatches in DACand ADC clock, drastically deteriorates the achievable floor. Incontrast, the proposed windowed PSD estimation is insensitiveto window-length mismatch.

2) Swept-Tuned (ST) Approach: Swept-tuned analyzerstune into different analysis frequencies by sweeping the localoscillator frequency of the down-mixer, from a start frequencyfsw,a to a stop frequency fsw,b during a sweep time Tsw.This sweep is used to mix the input signal down to anintermediate frequency (IF). At time t, the signal frequencymixed to IF, is fsw(t) = fsw,a + fspan mod(t, Tsw)/Tsw,where fspan = fsw,b − fsw,a. The center frequency isfsw,c = (fsw,a + fsw,b)/2. After downmixing to IF, thesignal is filtered with a band-pass filter (BPF) centered atIF. This BPF determines the resolution bandwidth (RBW)of the spectral analysis. The power of the BPF output ismeasured using an envelope detector, whose output is furthersmoothed by a so-called video bandwidth (VBW) filter. Swept-tuned analyzers typically offer several detector modes. Tominimize bias, an RMS detector should be used not only whenmeasuring absolute power [26] but also when making powerratio measurements [27].

By sweeping the whole bandwidth of the signal with a fineRBW, it is possible to get a PSD estimate similar to the TDmethod. However, such a measurement can take a significantamount of time, e.g., 10 seconds, since the sweep time mustbe high for high values of fspan and low values of RBW.More precisely,

Tsw = Cfspan

RBW2 , (8)

where C is a dimensionless constant [28]. By tuning into eachstop band separately, we can achieve faster measurements: Toobtain the average stop-band PSD Sy(M) in the numerator of(2), we make NM separate sweeps and set the m-th sweep’sstart and stop frequency to the m-th stop-band’s start and stopfrequencies, i.e., ωa,m2π and ωb,m

2π , respectively. The RBW mustbe small enough to prevent including power in the slope ofneighboring present-bands. In our WLAN application we usean RBW of 10 kHz, C = 10, and fspan = 312.5 kHz.

To obtain the average present-band PSD Sy(P), we couldproceed similarly as for Sy(M) above and make NP separatemeasurements. However, we again advocate for an approachdecreasing the measurement time: We make a single sweepover the whole bandwidth of the signal, since the contributionof the missing bands Sy(M) to Sy(P) is negligible for typicalinband error values of interest. For instance, if the true inbanderror is ≤ −20 dB, we obtain −20.04 dB = 10 lg(0.01/(1 +0.01)). Since the stopbands do not have to be resolved here,the RBW can be relatively high. We use fspan

RBW = 200, yieldingRBW = 200 kHz for a signal with a bandwidth of fspan =40 MHz. With C = 100 we still get Tsw in the order of100 ms. Though the power is integrated over the full signalbandwidth fspan, it is important to normalize by the present-bandwidth λ(P) when computing Sy(P).

IV. TRANSCEIVER IMPAIRMENTS AND THEIR INFLUENCEON EPR AND EVM

In the following, we review common analog transceiverimpairments and discuss their influence on the EPR and EVM.

6

EVM = -SNR = 10log10( 7Sv(P)= 7Sx(P))-90 -80 -70 -60 -50 -40 -30 -20 -10 0E

PR

-EV

M(d

B)

-3

0

3RM =80 dBRM =90 dB

Fig. 4. Simulated EPR estimation error in case of additive noise for twodifferent stopband rejection factors RM. At high SNR (−90 dB EVM), theEPR overestimates the EVM, because of the bias due to the limited stopbandrejection. At low SNR (0 dB EVM) the EPR underestimates the EVM becausewe defined the EPR as an error to signal plus error ratio.

A. Additive Noise

Consider the following signal model

y(t) = x(t) + v(t) , (9)

where x(t) = h(t) ∗ s(t), s(t) denotes the signal, h(t) is animpulse response, and ∗ denotes convolution. Assuming thenoise v(t) to be uncorrelated with x(t), the PSD of y(t) is

Sy(ω) = Sx(ω) + Sv(ω) . (10)

Inserting in the definition of the EPR in (2) yields

EPR{y(t)} =Sy(M)

Sy(P)(11)

=Sx(M) + Sv(M)

Sx(P) + Sv(P)(12)

Assuming an EPR test signal x(t), (6) holds, i.e., Sx(M) =εSx(P). If we furthermore assume Sv(M) = Sv(P) , i.e., thenoise power averaged over the stopband is representative forthe noise at the present bands, (12) becomes

EPR{y(t)} =εSx(P) + Sv(P)

Sx(P) + Sv(P)=εξ + 1

ξ + 1≈ 1

ξ, (13)

where ξ = Sx(P)/Sv(P). The approximation in (13) holdsfor ε � ξ and ξ � 1. With SNR = 10 log10 (ξ), RM =−10 log10 (ε), and EVM = −SNR as derived in AppendixA-C, we obtain the result depicted in Fig. 4.

Note that the fixed, systematic bias at high EVMs resultsfrom the definition of EPR as an error to signal plus errorratio and could be easily removed, whereas the bias at lowEVM could be decreased by increasing the stopband rejectionRM. In most practically relevant cases, however, covering therange between −70 and −10 dB EVM is sufficient and biascorrection is not necessary.

The PSD of the source signal s(t) filtered with h(t) isSx(ω) = |H(ω)|2Ss(ω). If the filter preserves the power ofthe communication signal s(t), the SNR is not affected by h(t)and we have Sx(P) = Ss(P). Because of the integral natureof S and the EPR, the exact frequency dependent behavior ofH(ω) is irrelevant to the EPR. The EPR behaves like data-aided EVM with a perfect equalizer.

B. Nonlinearity

In case of nonlinear DUT, e.g., a power amplifier (PA), thedistorted DUT output can be decomposed into a correlatedcomponent and an additive uncorrelated distortion noise com-ponent [9]. The major difference to the additive noise scenarioabove is that the distortion noise is not independent fromthe excitation signal x(t), i.e., the excitation signal affectsthe effective inband SNR, in general. However, if the EPRtest signal has similar amplitude statistics like the EVM testsignal, the effective inband SNR is the same in an EVMand EPR measurement, and consequently the EPR is able toestimate EVM for nonlinear DUTs. To make this more clear,we investigate a third-order polynomial system

yRF(t) = c1xRF(t) + c3x3RF(t) . (14)

Assuming a Gaussian input signal xRF(t), the autocorrelationfunction of (14) is given as [10]

Ry(τ) = (c1 + 3c3Rx(0))2Rx(τ) + 6c23R3x(τ) . (15)

Consequently, the PSD of the output signal is

Sy(ω) =(c21 + 6c1c3σ2x + 9c23σ

4x)Sx(ω) + 6c23Sx,3∗(ω), (16)

with the three-fold convolution term

Sx,3∗(ω) = (Sx ∗ Sx ∗ Sx)(ω) , (17)

and the variance of the input signal σ2x = Rx(0) (a zero

mean distribution of x(t) is presumed for this abbreviation).In OFDM receivers, in each frame, new equalizer coefficientsare estimated from the preamble. If the source signal and thenonlinear system are stationary within one frame, which isa reasonable assumption for OFDM and power amplifiers,the first three terms in (16) cannot be distinguished. Con-sequently, the whole gain factor c21 + 6c1c3σ

2x + 9c23σ

4x gets

equalized. Hence, only the uncorrelated distortion 6c23Sx,3∗(ω)contributes to the EVM.

By inserting (16) in the definition of EPR in (2), we seethat the EPR only captures the last term due to the missingexcitation at the stop-bands Sx(ω)|ω∈M ≈ 0. This rejection ofcorrelated distortion is a welcome feature, because it resemblesthe effect of using an equalizer in EVM tests, as discussedabove. If we have a system with higher order, we get additionalhigher-order convolution terms, but the principle remains thesame. Also, the extension to a Wiener-Hammerstein model,i.e., a static nonlinearity between two linear filters introducingmemory, is straightforward: Sy(ω) = F {Sx(ω)} in (16)becomes

Sy(ω) = |Hpost(ω)|2F{|Hpre(ω)|2Sx(ω)

}. (18)

Although the input and the output of the nonlinearity arefrequency-weighed, it is clear that (18) can again be decom-posed into a correlated term and an uncorrelated term. Thisholds even for more complicated nonlinearites with memory,as explained by the Bussgang theorem [9].

The EPR is completely blind to the correlated term includ-ing the correlated distortion, e.g., 6c1c3σ

2x + 9c23σ

4x in (16)

that arises if we have a non-zero third-order coefficient c3.A linear equalizer can compensate the correlated distortion,

7

1

Frequency (∆f)

Frequency (MHz)

dBdB

Sx,EPR(f)

Sy,EPR(f)Sy,EVM(f)

Sy,NPR(f)

without EQEVM

with EQEVM

Mask

EPR

−40

−60

−80

−15

−25

−28 −21 −7 7 21 28

−30

−30

−20

−20

−20 −10

0

0

0 10 20 30

Fig. 5. Wiener PA model simulation. Top plot: zoomed version of bottom plotwith abscissa in multiples of the OFDM tonespacing ∆f . At the stop-bandcenter frequencies ωc,M = {−21,−7, 7, 21} · 2π∆f , the EPR resemblesthe EVM with equalizer (EQ) at these tones.

since it is, for each OFDM frame, just a weighting factor thatis frequency-dependent for nonlinearities with memory. Thenonlinear nature of the weighting factor is apparent in (16)because it depends on the test signal power σ2

x. This is howevernot a problem for frame-based equalization because σ2

x isconstant within an OFDM frame. To have an EPR resemblingthe EVM, the EVM analyzer’s equalizer must be able tocorrect the correlated distortion sufficiently. With OFDM, thisis typically the case if the nonlinear memory effects are shorterthan the cyclic prefix length.

Another important issue with nonlinearities that have mem-ory is that the error (the uncorrelated distortion) is, in general,frequency-dependent. To estimate EVM correctly, the errorobserved in the EPR stopbands must be representative forthe error of the whole inband, as we also required for (13).The individual stopband EPRs resemble the EVM at these fre-quencies which is illustrated in Fig. 5. The overall EPR is themean over the individual stopband EPRs as defined in (5). Theoverall EVM is the mean over the EVM at all data subcarriers.If the mean over the EVM at the stopband subcarriers differsfrom the overall EVM, the single-measurement multi-stopbandEPR will also differ from the overall EVM.

In practice, most transceiver DUTs feature an error thatis well-behaved, i.e., not changing abruptly over the inbandfrequencies. If, however, there is reason to expect highlyfrequency-dependent error, e.g., when a dynamic elementmatching DAC [29] is part of the DUT, there is the possibilityof making several measurements with different stop-bandfrequencies.

Fig. 5 illustrates the EPR for the Wiener PA model from[30]. This model consists of an FIR filter with coefficients[0.7692, 0.1538, 0.0769] followed by a quasi-memoryless [31]nonlinear model [32] given as

y[n] =1.1x[n]

1 + 0.3 |x[n]|2exp

j0.8 |x[n]|2

1 + 3 |x[n]|2. (19)

In the simulation belonging to Fig. 5, the EVM withoutequalizer is −16.2 dB, whereas it is −32.7 dB with equalizer.The EPR is also −32.7 dB, i.e., it equals the EVM withequalizer.

Note that (14) is a continuous-time passband model,

1

Frequency (∆f)

dBdB

PSD Sx(f)EPR Input

PSD Sy(f)EPR Output

de-rotationEVM with

de-rotationEVM without

−40

−50

−50

−60

−70

−100−28 −3 3 28

−25.5 −25

−25

−24.5 −2 0

0

0

2 24.8 25

25

25.2

Fig. 6. Phase noise simulation illustrating the importance of the stopband-width BWstop for EVM estimation. Bottom: EPR test signal PSD, DUToutput PSD, and EVM per subcarrier. Top: Zoomed missing bands. Left: WithBWstop = ∆f , i.e., the missing bandwidth equals the OFDM tonespacing,good estimates of EVM with de-rotation (phase tracking) can be expected.Middle: With BWstop = 6∆f , the EPR underestimates the EVM. Right:With BWstop = 0.1∆f , the EPR overestimates the EVM with de-rotation.

whereas the equalizer is typically estimated from discrete-timebaseband data. Still, the above argumentation on the EVM andthe equalizer is valid, because when mapping from passbandto baseband, the polynomial structure is obtained. Only thecoefficient values get scaled with factors depending on theirorder [31], [33]. The autocorrelation for a baseband third-ordernonlinearity can be found in (5.37) in [9].

C. Phase Noise

Phase noise occurs due to jitter of the local oscillator (LO)of the mixer. A baseband model for phase noise is

y(t) = x(t)eφ(t) ≈ x(t) + x(t)φ(t) . (20)

where φ(t) is the phase noise with PSD Sφ(ω). The first-order Taylor series approximation in (20) is valid if φ(t) <<1, which is a reasonable assumption for LOs in moderntransceivers [3]. The PSD of y(t) is hence

Sy(ω) ≈ Sx(ω) + (Sx ∗ Sφ) (ω) . (21)

The additive error term Se,PN (ω) = (Sx ∗ Sφ) (ω) is statis-tically dependent on Sx(ω). A double-side-band PLL phasenoise model L(f) =

√2Sφ(2πf) is given as [3]

L(f) =B2

PLLL0

B2PLL + f2

(1 +

fcorner

f+ Lfloor

), (22)

where f > 0 denotes the frequency offset from the carrier,BPLL is the PLL −3 dB bandwidth and L0 is the inband phasenoise level in rad2/Hz, fcorner is the flicker corner frequency,and Lfloor the noise floor. Depending on the bandwidth ofSφ(ω) compared to the OFDM subcarrier spacing ∆f , twocases can be distinguished: Inter-carrier interference (ICI)prevails if BWPLL > ∆f/2 and common phase error (CPE) isdominant if BWPLL < ∆f/2. While it is very hard to removeICI in a receiver, CPE can be mitigated by phase tracking[3]. Phase tracking is mandatory for EVM measurementsaccording to the 802.11ac WLAN standard [1].

With EPR, the convolution in (21) spills power fromneighboring present bands into the stop-band. Since CPE can

8

1

Frequency (∆f)

dB

Sx(f)Input PSDafter IQ-MM

PSD Sy(f)

per subcarrierEVM

−28 −21 −7 7 23 28

−80

−60

−40

−20

0

0

Fig. 7. IQ mismatch simulation illustrating the importance of the missing bandlocations ωc,m.With the symmetric tones at ±7∆f , IQ mismatch is excluded.With asymmetric locations (−21,+23)∆f , IQ mismatch is included.

be corrected in EVM, we do not want CPE in our stop-band, contributing to the error power in the numerator of (2).Similarly, we want ICI to contribute to the stop-band powerbecause it is also included in EVM. Since what is consideredICI and what is CPE depends solely on ∆f for a given phasenoise bandwidth, it is clear that also our stop-band widthBWstop must be related to ∆f . As can be seen in Fig. 6,using BWstop = ∆f works fine for estimating the EVM withactivated phase tracking. For Fig. 6, we used rather narrow-band phase-noise with BPLL = 10 kHz, fcorner = 0.5 kHz,Lfloor = −150 dB, and L0 = −90 dB in order to clearlysee the differences between EVM with and without phasetracking and the influence of different stop-band widths. Asshown in the measurement chapter V, BWstop = ∆f ishowever also appropriate for higher phase noise bandwidths,e.g., BPLL = 160 kHz.

D. IQ Mismatch

A baseband model for transmitter IQ mismatch is [34]

y(t) = (x ∗ g1)(t) + (x∗ ∗ g2)(t) , (23)

i.e., x(t) = xI(t) + xQ(t) and its conjugate x∗(t) areconvolved with impulse responses (IRs) given as

g1(t) =(hI(t) + geϕhQ(t)

)/2 , (24a)

g2(t) =(hI(t)− geϕhQ(t)

)/2 . (24b)

Here, g is the mixer amplitude imbalance factor, ϕ the phaseimbalance, and hI(t) and hQ(t) the IRs of the the in-phaseand quadrature path of the D/A converter, respectively. Thepower spectral density (PSD) of (23) is

Sy(ω) = |G1(ω)|2 Sx(ω) + |G2(ω)|2 Sx(−ω) , (25)

where |Gi(ω)| is the Fourier transform of gi(t), with i ∈{1, 2}. If |G2(ω)| > 0, there is IQ mismatch, and an exci-tation at ω0 causes interference at the mirrored frequency −ω0.

With EPR, we can choose where to place the stop-bands.Placing them symmetrically around ω = 0, excludes IQmismatch, because then there is no excitation at mirroredfrequencies. If, however, the stop-bands are chosen to beasymmetric, the effect of IQ mismatch is included in the EPRresult. In practice this can be very helpful when trying to sort

out whether IQ imbalance is the limiting factor that determinesthe inband-error. By contrast, this is not possible with EVM,since EVM does not allow for excluding IQ mismatch. InFig. 7, we illustrate the EVM and EPR for an IQ mismatchsimulation. The used IQ mismatch model is a discrete-timeequivalent of (23) with

hI [n] = 0.98δ[n] + 0.02δ[n− 1] (26a)hQ[n] = 0.94δ[n] + 0.06δ[n− 1] (26b)geϕ = 1.01 · exp (π2π/360) (26c)

where δ[n] is the delta impulse sequence, i.e., we use two-tap FIR filters to model frequency-dependent mismatch of theDAC’s I- and Q-path, a gain imbalance factor g = 1.01 andπ degree phase imbalance.

In Fig. 7, we use an EPR test signal Sx(f) featuring fourstop-bands. Two of these stop-bands (±7∆f ) are symmetricaround zero, whereas the other two are at (−21,+23)∆f , i.e.,they are asymmetric and do not have an equivalent mirroredaround zero. The simulation confirms our analysis above:The symmetric stop-bands are blind to IQ mismatch, i.e.,the output PSD resembles the input PSD at these bands.If we only use symmetric bands to compute the EPR, weget nearly −90 dB, i.e., the best our analysis window canachieve, although there is strong IQ mismatch. The asymmetricbands (−21,+23)∆f , on the other hand, accurately sample thefrequency-dependent error, depicted as EVM per subcarrier.Using only asymmetric stop-bands allows to estimate EVM ifIQ mismatch is considered as a part of the DUT. Using bothsymmetric and asymmetric locations at the same time as inFig. 7, allows for checking whether the error is dominated byIQ mismatch by just looking at the PSD.

As also discussed in Sec. IV-B, for frequency-dependentinband error, the mean over the error at the stopbands mustapproach the overall error. If this is fulfilled the configurationof the asymmetric stopband is not crucial as long as there isno attenuation of the test signal PSD at mirror-frequencies ofthe EPR integration range. In Fig. 7, we used (−21, 23)∆f ,i.e. a stopband offset of 2∆f to achieve asymmetry. Usingonly 1∆f would not suffice, because the stopband transitionwould create attenuation at a mirror frequency.

V. MEASUREMENT RESULTS

To verify the proposed EPR measurement method experi-mentally, we made measurements comparing EVM with EPR.Our measurement setup is depicted in Fig. 8. We use a PCconnected to the Agilent MXG signal generator and an R&SFSQ Analyzer via LAN. The FSQ’s 10 MHz sync output isconnected to the MXG’s 10 MHz sync input.

A. Signal Generation

We generate the 802.11ac source signal s[n] using a PC run-ning MATLAB and the VHT waveform generator tool cited inAnnex S of [1]. Unless otherwise stated, we use a single burstwith 320 data symbols, 40 MHz bandwidth, and modulationcoding scheme (MCS) 1, i.e., QPSK subcarrier modulation.Upsampling to a sample rate of 160 MHz is achieved by zero-padding in the IFFT. For generating EPR excitation signals,

9

Fig. 8. Measurement setup including a PC with MATLAB, Agilent MXGsignal generator, and R&S FSQ analyzer. As nonlinear DUT we use an RFMDRFPA5522 power amplifier (PA) with a 10 dB attenuator at its output.

we filter ˜s[n] according to section III-C. We use the standardpilot tone locations {−53,−25,−11,+11,+25,+53} for ourcenter frequencies ωc,m to obtain symmetric stop-bands. Forasymmetric locations, we offset the positive tones by −2, i.e.,we use {−53,−25,−11,+9,+23,+51}. After downloadingthe signal x[n] to the MXG, the MXG converts the digitalbaseband signal to an analog RF signal, centered at 5.6 GHz.

B. Device Under Test

As nonlinear DUT, we use an RFMD RFPA 5522, acommercially available, three stage power amplifier (PA)for 802.11a/n/ac applications. To operate the PA in a widerange, from a linear regime to deep saturation, we sweepthrough the following 15 MXG analog output gain values:{−30,−25,−20,−15,−12.5,−10,−9, . . . ,−1} dB. To testthe effect of IQ mismatch and phase noise on EVM and EPR,we add IQ-mismatch and phase noise to the baseband signalin MATLAB, using the models discussed in Sec. IV. TheIQ-mismatch model is given in (26). For the phase noiseprofile in (22), we used the parameters BPLL = 160 kHz,fcorner = 2 kHz, Lfloor = −170 dB, and L0 = −93 dB.To get the lowest measurable inband error for reference, wemade measurements connecting the MXG’s RF output directlyto the FSQ’s RF input via an RF cable. However, we were ableto measure the lowest inband error by including the PA andthe 10 -dB attenuator in the measurement chain and drive itat a low level (around 2 dBm), because the PA is linear atlow levels. The resulting reference (best) RF chain results aresummarized in Table I and correspond to the leftmost datapoints of the PA power sweep depicted in Fig. 9.

C. EVM and EPR Analysis Methods

We use the FSQ analyzer in three different measurementmodes, depending on the EVM or EPR measurement ap-proach, as described in the following. To have a referenceto compare our own EVM analyzer code implemented inMATLAB, we used the K91ac EVM analyzer firmwarerunning on the FSQ. We use phase tracking, and makechannel estimation on preamble and data. We refer to theresulting average EVM over all data subcarriers as EVMFSQ.Furthermore, we are interested in the EVM at the pilot tones

EVMFSQ,pilots, since our EPR stop-bands are also at the pilotlocations, and we expect the FSQ’s phase-tracking algorithmto achieve the lowest EVM at the pilot-tones.

Our own, data-aided EVM analyzer implemented inMATLAB analyzes time-domain (TD) baseband data, hencewe refer to it as EVMTD. In contrast to the FSQ’s proprietaryanalyzer, we can use our EVM analyzer in simulations, andhave full understanding of the processing. In measurements,we obtain the baseband data via the FSQ’s I/Q mode datatransfer functionality, i.e., the FSQ downconverts the signalfrom 5.6 GHz to baseband and analog to digital conversion.

For EPRTD, i.e., the EPR based on PSD estimation usingTD data, the measurement chain is exactly the same asfor EVMTD, only the excitation signal and the analysis isdifferent. Having the same measurement chain for EPRTD andEVMTD is beneficial when directly comparing the results.

D. Calibration

The FSQ’s input attenuator (ATT) and vertical reference(VREF) settings are crucial for obtaining best (lowest) andcomparable results. Particularly at EVMs below −30 dB, weobserved that the FSQ’s automatic ATT and VREF selectiondelivers results several dB worse than those obtained withmanually optimized settings. Furthermore, the best settings arenot necessarily the same in the different modes (WLAN, TD-IQ, swept-tuned channel power measurement). Accordingly,for each tested power level, we optimized the ATT and VREFsetting to achieve the lowest EVM or EPR, for each method.Since the signal path is exactly the same for EPRTD andEVMTD, also the optimal ATT and VREF settings are thesame.

The generator and the analyzer we used receive yearlycalibration by the manufacturer. Apart from the selection ofthe ATT and VREF setting as described above, user calibrationis not very critical, because we are using a signal analyzer andnot a network analyzer. Furthermore, we are only interested incomparing different methods (EVM, EPR) and not so much inthe absolute accuracy, e.g., regarding the power measurement.The different analysis methods (EVM, EPR) themselves donot need absolute power, but power ratios. Consequently,systematic errors in the absolute power level are excludedfrom the result in the ordinate of Fig. 9 by design. Theabsolute power measurements varied up to 0.3 dB between thedifferent analysis methods. We removed this variation fromour results by using the same data (the power measured withthe EVMTD) for the abscissa of all the different methods inFig. 9. This is valid, because of the following: All excitationsignals are scaled digitally to have exactly the same power.The measurement chain (generator, cables, PA) is always thesame and differences in output power due to (slightly) differentsignal statistics (e.g., PAPR) are negligible. Consequently, itis reasonable that the power at the output of the DUT is thesame for all analysis methods.

E. Results

The results of the PA power sweep are depicted in Fig. 9.The wide range of tested EVM conditions can be seen from

10

(a) (c)

(b) (d)

Inba

ndE

rror

(dB

)D

evia

tion

(dB

)

PA Output Power (dBm)

PA Output Power (dBm)

PA Output Power (dBm)

PA Output Power (dBm)

EVMFSQ

EVMFSQ,pilots

EVMTD

EPRTD,asymm

EPRST,asymm

EPRTD,symm

EPRST,symm

includes IQ-mismatchlike EVM, asymmetric EPR

excludes IQ-mismatchsymmetric EPR

3

1

0

0

−0.5

−1

−1

−1.5−2

−2−3

−10 −10

−20 −20

−30 −30

−40 −40

−50 −50

−60 −60

2

2

2

2

2

4

4

4

4

6

6

6

6

8

8

8

8

10

10

10

10

12

12

12

12

14

14

14

14

16

16

16

16

18

18

18

18

20

20

20

20

22

22

22

22

24

24

24

24

26

26

26

26

28

28

28

28

30

30

30

30

Fig. 9. Inband error (IBE) measurement results comparing several methods/settings for EVM and EPR. The DUT is a WLAN power amplifier (PA). Thesignal generator’s output gain is swept. (a) PA measurement. (b) Same data as (a), but deviation from EVMFSQ is shown. (c) PA measurement with IQmismatch added to the test signal x[n]. (d) Same data as c), but deviation from EVMFSQ is shown.

Fig. 9 (a). At the highest level (around 30 dBm), the PA isin saturation and the distortion leads to −15 dB of inbanderror. At low levels, the PA behaves very linear, achievingEVMs down to −53 dB. In essence, the results of all methodsagree over the whole power range. To see differences in moredetail, the deviation from EVMFSQ is plotted in Fig. 9 (b).EPRTD and EVMTD have an absolute difference less than0.35 dB over the whole test range. Since the EPR obtained viaswept-tuned (ST) RF power measurements allows for a lowermeasurement floor, the deviation defined EPRST,deviation =EPRST−EVMFSQ is negative for low PA output powers. InFig. 9 (c), we present results of the same PA power sweepas in (a) but with IQ mismatch added to the test signalx[n]. With symmetric stopbands, the EPR does not captureIQ mismatch. Therefore, EPRsymm in Fig. 9 (c) agrees withthe result without added IQ mismatch in subplot (a). Withasymmetric rejection bands, IQ mismatch is included andhence EPRasymm resembles the EVM in Fig. 9 (c).

Table I presents inband error results of the RF chain pre-sented in Sec. V-B, i.e., at low PA output power (2 dBm). Thevalues of EVMTD and EPRTD equal, which indicates that ifthe measurement chain is the same, the EPR is able to estimatethe EVM very well. With the swept-tuned method EPRST alower noise-floor of −56.3 dB can be achieved. In the caseof added phase noise shown in Table II, EPRST resemblesEVMSIM and EVMFSQ,pilots closely. As expected, the EVMresult at the data tones is slightly (0.6 dB) worse. EVMTD andEPRTD are slightly biased to higher values by the higher floor

TABLE IRF CHAIN: MEASURED INBAND ERROR (DB)

EVMFSQ EPRST EVMTD EPRTD

-51.5 -56.3 -47.2 -47.2

TABLE IIPHASE NOISE: MEASURED INBAND ERROR (DB)

Method EVM EPR

FSQ/ST -38.4 -38.9TD -38.1 -38.5SIM -38.9 -39.0

FSQ,pilots -39.0

TABLE IIIIQ MISMATCH: MEASURED INBAND ERROR (DB)

Method EVM EPRasymm EPRsymm

FSQ/ST -35.8 -35.9 -56.3TD -35.8 -35.6 -46.8SIM -35.8 -35.7 -88.6

of the TD RF signal path. The results in Table III confirm ourfindings in Sec. IV regarding IQ mismatch. With asymmetricstopbands, however, the EPR results resemble the EVM resultsfor IQ mismatch. Symmetric stopbands, however, lead to EPRresults resembling the case without added impairment.

11

VI. DISCUSSION

A. Measurement Uncertainty

In the following, we discuss the uncertainty involved in EPRmeasurements. Uncertainty is the doubt about the measure-ment result [35], i.e., a measure of the variation to expect whenmaking repeated measurements affected by random errors.Uncertainty analysis of EVM measurements is discussed in[7], [36], and references [20-29] in [7]. We discuss systematicerrors and bias, i.e., the expected deviation from the true value,separately in Sec. VI-B. We define three classes depending onthe source of uncertainty. These are

1) Randomness of the excitation signal.2) Randomness of the DUT.3) Randomness of the remaining measurement chain.

All three types of uncertainty can affect the repeatability ofthe measurement. If we use a fixed test signal, which we doin our measurements, there is no uncertainty of the first type.However, there can be bias, as discussed in Sec. VI-B.

Uncertainty of the second type arises due to limited ob-servation (time and bandwidth) of random DUT errors, e.g.,thermal noise of a PA. The EPR has more type-2 uncertaintythan EVM, because the EPR observes the error only in afraction αBW = λ(M)

BW < 1 of the bandwidth BW usedfor averaging the error in EVM. If there is only type-2uncertainty, the variance of EPR is hence 1/αBW > 1 timesthe variance of EVM, if the measurement duration is the same.For deterministic error, e.g., a nonlinear DUT and negligibletype-1 and type-3 uncertainties, repeated measurements havethe same result, so both EPR and EVM have zero variance.

The type-3 uncertainties of EVMTD and EPRTD are thesame, because the measurement chain is the same. Althoughwe do not know the internals of the FSQ and the EVManalyzer software, we expect the EVMFSQ to have similaruncertainty as EVMTD. For EPRST, a different detector(RMS instead of sample) is used and random errors in the RXIQ path are excluded. Still, our experience is that the differencein uncertainty of EPRST compared to EPRTD is small, inpractice. This is also supported by the informal experimentdescribed below. Apart from excluding synchronization errorspotentially biasing the result, the swept-tuned principle allowsfor EPR measurements with lower measurement floor at highbandwidths.

To get more insight for the amount of type-3 uncertaintiesand the repeatability of our measurements, we made tensuccessive trials of the power sweeps in Fig. 9 with fixedexcitation signals. The deviation from the mean (in dB) waslow for all methods, over the whole power range. For the EPRmethods the deviation was within ±0.17 dB over the wholerange. For EVM it was in the same range (±0.12 dB) apart foran outlier deviating up to −0.32 dB for higher levels. At lowerlevels, the uncertainty of the EPR measurements was slightlyhigher compared to EVM. This is reasonable because at lowlevels, the influence of additive noise is stronger and hencewe have type-2 uncertainty, leading to an increased varianceof EPR compared to EVM.

B. Bias and Stopband Selection

In the following, we discuss bias, i.e., the systematicdeviation of the EPR from the true EVM value. Since, ingeneral, the bias depends on the DUT, it is difficult toremove. However, bias can often be avoided by setting upthe EPR measurement right. Below, we discuss both sides ofthe following tradeoff. Using only few stopbands is beneficialin order that the same error occurs in response to the EPRtest signal as in response to the EVM test signal. However,for heavily-frequency-dependent error it is important that weobserve the error at many different frequency-locations, i.e.,use many stopbands. Our experience is that using multiplestopbands, each one subcarrier spacing broad, with an overallstopband width of 3 to 10% of the original signal bandwidth isa very good compromise, in practice (six stopbands at 40-MHzWLAN correspond to about 5%). Choosing the width of eachstopband to be the OFDM subcarrier spacing is important toavoid bias in case of phase noise. Placing the stopband-centersat OFDM subcarrier-frequencies is convenient for comparisonwith subcarrier-dependent EVM. Then, the individual EPRsat each stopband resemble the EVMs at these subcarriers, i.e.,EPR(km) ≈ EVM(km). For nonlinear DUTs this is, however,only true if the EPR test signal statistics are sufficiently similarto the EVM test signal statics. While for static nonlinearitiespreserving the amplitude statistics (the PDF) is sufficient, fornonlinearities with memory it can be important to also preservethe correlation and higher order moments of the test signal.However, retaining the PDF, e.g., a CSCN distribution, doesnot mean that the EPR test signal has the same PSD as thesource signal (and consequently also not the same autocor-relation). Quite the contrary, it cannot have the same PSD,since we need a PSD with stopbands for EPR, whereas theEVM test signal features no stopbands. The more bandwidthwe null for EPR, the less will the PSDs resemble, and the morewe are risking to produce a different amount of distortion atthe output compared to the EVM signal which may lead to abiased EPR. However, we observed in many experiments withmeasured and simulated systems, that this issue is not verycritical in practice when nulling, e.g., 10% of the bandwidth.A related issue is that the test signal should not be too short inorder to excite the DUT in the same way as a standard EVMtest signal. We used 320 symbols according to the EVM testWLAN standard, i.e., 1.28 ms.

Now we discuss the issue of observing the error adequately,i.e., the observed error must be representative for the overallerror. When the frequency-dependence of the error is mild(which is the case for most practical systems) the abovepresented stop-band placement works well for estimating theEVM. Then the exact number (between 3 or 8% of the numberof number of OFDM subcarriers) and absolute location of thestopbands is also not critical. We proposed to use to use thepilot locations as stopband-centers, because they are easy toremember and the result can be directly compared to EVMaveraged at the pilots, which is provided by most commercialEVM analyzers. Sometimes, the error increases or decreasesat the band-edges or around DC, so it can be useful to includethe outermost and innermost modulated subcarriers. Having a

12

look on the individual EPR values is always advisable. If theEPR varies a lot, it is sensible to make a second measurementwith different stopband locations. If the result differs fromthe first one, it makes sense to make several measurements,sweeping the stopband locations over the whole inband.

VII. CONCLUSION

We have presented an NPR measurement method with thegoal of estimating EVM. To discern our method from thetraditional approach with a single, broad notch in the center ofthe spectrum, and to account for the fact that NPR is inverselydefined, we introduced the term error power ratio (EPR) forour NPR-based EVM estimation method. EPR is an attractivealternative to EVM because, in contrast to EVM, EPR does notneed high-accuracy demodulation and digitization of the wholesignal bandwidth. Rather, a standard swept-tuned spectrumanalyzer is all that is needed to make high accuracy, low-floormeasurements more or less independent of the bandwidth.

EPR can be a good estimate of EVM (with equalizerand phase de-rotation employed) in case of additive noise,nonlinearity, phase noise, and IQ mismatch. We have explainedthis by analytical considerations illustrated with exemplarysimulation results, and verified by measurements. Besides esti-mating EVM by using asymmetric stop-bands, EPR providesthe possibility to exclude IQ mismatch by using symmetricstopbands. This can be handy when tracking the source oflimited EVM performance or when measuring EVM of DUTswith very low EVM, without biasing the result with non-idealIQ-modulation and/or -demodulation.

We are convinced that the presented EPR method can bea valuable tool for RF engineers in the lab when trying toestimate the EVM performance of transceiver building blocks,such as power amplifiers. Although we focused on WLANsignals, our method should be readily applicable to otherOFDM-based communication standards. However, for signalsthat are not circularly symmetric normal (CSCN), some furtherwork is necessary. Straightforward linear filtering changes thestatistics of a non-CSCN signal, in general. Hence, a key issuewhen trying to estimate EVM for non-CSCN signals is how togenerate a signal retaining the communication signal statisticsbut featuring the required stop-bands.

ACKNOWLEDGMENT

The authors would like to thank Intel Connected HomeDivision (CHD), Villach, Austria, for their guidance, supportand access to laboratory equipment.

APPENDIX AOFDM AND EVM

A. Orthogonal Frequency Division Multiplexing (OFDM)

A baseband OFDM signal s(t) : R→ C can be written as

s(t) =∑l∈L

sl(t− lT ) , (27)

where T = Tw + Tg + TK is the symbol duration consistingof guard Tg , window Tw and effective symbol time TK [21].The symbol index set L is a connected set of integers. Unless

otherwise stated, we assume L = {0, . . . , NL − 1}. The lthOFDM symbol is given as

sl(t) = w(t)∑k∈K

S[l, k]e 2πkTK

t, (28)

where w(t) is a window function with finite support −Tw −Tg ≤ t ≤ TK +Tw fulfilling

∑l∈Z w(t− lT ) = 1 [21]. S[l, k]

is a complex {data, pilot, null} symbol modulated on the kthsubcarrier of the lth OFDM symbol. The subcarriers (tones)K indexed by k ∈ K = {−NK/2, . . . , NK/2− 1}, maybe divided into disjoint subsets for data carrying, pilot, andunused (null) tones D, P , and U , respectively. The associatedcardinalities are ND, NP and NU , respectively, with NK =ND+NP +NU . In the 40-MHz WLAN standard [1], we haveNK = 128, ND = 108, NP = 6, P = {±53,±25,±11},U = {−64,±63,±62,±61,±60± 59,±1, 0}, NU = 14.

B. Error Vector Magnitude (EVM) Definition

The DUT output symbols are given as

Y [l, k] =1

TK

∫ lT+TK

t=lT

ys(t)e− 2πkTK

(t−lT )dt , (29)

where ys(t) indicates that the DUT output y(t) must besynchronized to s(t). For instance, a potential time-delay mustbe compensated, e.g., with the method discussed in [37]. Inpractice, an FFT is used to compute Y [l, k] from ys[n]. Theconstellation symbol error is defined

E[l, k] = Yc[l, k]−X[l, k] , (30)

where Yc[l, k] is a corrected version of Y [l, k]. In data-aidedEVM analysis, the subcarrier modulation symbols X[l, k]are the known, source symbols, i.e., X[l, k] = S[l, k]. Therequired processing to obtain Yc[l, k] from Y [l, k] correspondsto the signal enhancement facilities of a receiver: To removelinear filtering effects, a frequency-dependent equalizationfactor CEQ[k] ∈ C is introduced. Potential degradation dueto a common phase error (CPE) is combated using a symbol-dependent phase de-rotation factor CCPE[l] ∈ C. With that, theobserved constellation symbol enhanced by equalization andphase tracking is given as

Yc[l, k] = CEQ[k]CCPE[l]Y [l, k] . (31)

With data-aided analysis, i.e., the transmitted data symbolsX[l, k] are available in the analyzer and are used as a refer-ence, the equalizer and de-rotation coefficients can be foundby minimizing the error in a least squares sense:

CEQ,opt[k] = argminCEQ[k]

∑l∈L

|CEQ[k]Y [l, k] −X[l, k]|2

=

∑l∈LX[l, k]Y ∗[l, k]∑l∈L Y

∗[l, k]Y [l, k](32)

CCPE,opt[l] = argminCCPE[l]

∑k∈S

|CCPE[l]Yc,EQ[l, k]−X[l, k]|2

=

∑k∈S X[l, k]Y ∗c,EQ[l, k]∑

k∈S Y∗c,EQ[l, k]Yc,EQ[l, k]

(33)

13

with Yc,EQ = CEQ[k]Y [l, k], meaning that equalization andde-rotation is performed sequentially, in contrast to findingthe joint global optimum which would be much more involved.Furthermore, we implicitly assume already compensated time-delay and frequency offset, which allows for simple least-squares optimization of the compensation parameters, whereasthe joint determination of optimal compensation parametervalues is a non-convex problem [6]. The mean constellationpower over a set of subcarriers C with cardinality |C| is denoted

PSC =1

|C|∑k∈C

PS [k] , (34)

where the tone-dependent power PS [k] is obtained by averag-ing over all symbols, i.e.,

PS [k] =1

NL

∑l∈L

|S[l, k]|2 . (35)

The error vector power (EVP) is defined as the mean errorpower over all data tones D normalized by the respectivereference constellation power.

EVP =PED

PXD

(36)

EVM =√

EVP . (37)

If every constellation point occurs with the same probability,PXD is an estimate of the average constellation power, whichis a constant for a given modulation format. In practice, forNl ≥ 100, PXD is more or less identical to the averageconstellation power. In any case, we use the exact value ofPXD for computing the EVM. We see that EVP is definedas a power ratio. The difficulties and flaws of EVM comeinto play because for the error power PED , the demodulatedsymbols Yc[l, k] are needed as can be seen from (30). Conse-quently, accurate synchronization and equalization is required.To inspect the distribution of the EVM over the subcarriers k,we define

EVM[k] =√PE [k]/PXD . (38)

C. EVM for Additive Noise

Transforming the linear additive noise model in (9) toFrequency domain yields

Y (ω) =H(ω)X(ω) + V (ω) . (39)

Assuming that the impulse respones h(t) has a discrete timeequivalent h[n] with finite support0 < n < Nh − 1, whereNh ≤ Ng must be smaller than the effective OFDM guardinterval, the constellation symbol is given as (9)

Y [k, l] =H[k]X[k, l] + V [k, l] . (40)

where H[k] is the NK point DFT of h[n]. Inserting (40) in(30) gives the error

E[k, l] = Y [k, l]CEQ[k]−X[k, l] (41)= CEQ[k]H[k]X[k, l] + V [k, l]−X[k, l] (42)= V [k, l] . (43)

To have (43) follow from (42), it is assumed that CEQ[k] =1/H[k], i.e., the equalizer is able to correct the channel. Sincethe error contains only the noise, the data-aided EVM in dBequals -SNR averaged over the data bins, which is a wellknown result [38].

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Karl Freiberger was born in Graz, Austria, in 1984.He received the Dipl.-Ing. degree in electrical engi-neering and audio engineering from Graz Universityof Technology, Austria and the University of Musicand Performing Arts, Graz, Austria, in 2010. From2010 to 2013 he was with bct electronic GesmbH,Salzburg, Austria, working on acoustic echo cancel-lation, multichannel speech enhancement, and acous-tic design for communication terminals. From 2013to 2015 he was a researcher at the Telecommuni-cations Research Center Vienna (FTW), working on

digitally enhanced transceivers. Since 2015 he works as a researcher andlecturer at the Signal Processing and Speech Communication Laboratory, GrazUniversity of Technology, where he is currently pursuing a Ph.D. degree. Hisresearch interests cluster around signal processing, with a current focus ondigital pre-enhancement of WLAN transmitters and related optimization andmeasurement problems.

Harald Enzinger was born in Judenburg, Austria,in 1986. He received the Dipl.-Ing. (FH) degreein electronic engineering from the University ofApplied Sciences FH Joanneum in 2009 and theDipl.-Ing. degree in information and computer en-gineering from Graz University of Technology in2012. From 2012 to 2015 he worked as a reseracherat the Telecommunications Research Center Vienna(FTW). He is currently with the Signal Processingand Speech Communication Laboratory at Graz Uni-verity of Technology where he is working towards

a Ph.D degree. His research interests include behavioral modeling and digitalpredistortion of radio frequency power amplifiers and digital enhancement ofmixed-signal circuits.

Christian Vogel was born in Graz, Austria, in 1975.He received the Dipl.-Ing. degree in telematics, theDr. techn. degree in electrical engineering, and theVenia Docendi in analog and digital signal process-ing from Graz University of Technology, Austriain 2001, 2005, and 2013, respectively. From 2006to 2007 Dr. Vogel was Assistant Professor at GrazUniversity of Technology, from 2008 to 2009 he wasan Erwin Schrodinger postdoctoral research fellowat the Signal and Information Processing Laboratoryat ETH Zurich, Switzerland, and from 2010 to

2014 he was key researcher at the Telecommunications Research CenterVienna (FTW). He is currently head of the degree program electronics andcomputer engineering at FH JOANNEUM - University of Applied Sciences,Austria and Adjunct Professor at Graz University of Technology, Austria. Hisresearch interests include the theory and design of digital, analog, and mixed-signal processing systems with emphasis on communications systems anddigital enhancement techniques. He worked on all-digital phased-locked loops,transmitter architectures, digital predistortion, vehicular communications, GPSreceivers, and factor graphs, but is most known for his contributions tothe understanding and further development of time-interleaved ADCs. Dr.Vogel is author and co-author of more than 70 international journal andconference papers, holds several international patents, and is the co-authorof five papers that have received best paper awards including the 2009 IEEECircuits and Systems Society Outstanding Young Author Award and the 2014IEEE Circuits and Systems Darlington Best Paper Award.