lecture 4a - papenlab1.ucsd.edupapenlab1.ucsd.edu/~coursework/wes/slides/lec4a-matchedfilter.pdf ·...

Lecture 4A

MatchedFilterReciever

MatchedFilters forBandlimitedChannelsIntegrate andDumpReceiver

Bit ErrorRate

Eb/N0Energy Eb

Noise DensityN0

Single sample(nointegrator)

Integrator(matchedfilter)

Lecture 4ABasic BPSK Modem

Wireless Embedded Systems - Communication Systems Lab - Fall 2017 Lecture 4A 1

Lecture 4A



Bit ErrorRate

Eb/N0Energy Eb

Noise DensityN0



Matched Filter

Want to determine the filter that maximizes the value of ζ given by

ζ =r1 − r0

2σ , (1)

where r is the sampled value after the detection filter and σ is the valueof the noise distribution after the filter.To determine the form of the optimal filter function, it is useful tomaximize ζ2

ζ2 =(s1 − s0

2

)2 r2

σ2 , (2)

where r1 = s1r and r0 = s0r where s1 and s0 are the symbol valuesfrom the constellation.Value r is the sampled value for a unit amplitude symbol value.The constant (S1 − S0)/2 in (2) does not depend on the filter.


Lecture 4A



Bit ErrorRate

Eb/N0Energy Eb

Noise DensityN0



Schwarz Inequality

To maximize ζ, it is sufficient to maximize

r2

σ2 =

(∫ ∞−∞

p(t)f(T − t)dt)2

N02

∫ ∞−∞

f2(T − t)dt, (3)

The Schwarz inequality applied to p(t) and f(T − t), states that(∫ ∞−∞

p(t)f(T − t)dt)2

≤∫ ∞−∞

p2(t)dt∫ ∞−∞

f2(T − t)dt, (4)

with equality if f(T − t) equals p(t).This is often written as f(t) = p(−t) with T set to zero. Referring to (4), ζ ismaximized if the received electrical pulse p(t) is correlated with itself.


Lecture 4A



Bit ErrorRate

Eb/N0Energy Eb

Noise DensityN0



Matched Filter

For a complex-valued received pulse, the detection filter f(t) is thecomplex conjugate p∗(t) of the received pulse. The correspondingimpulse response of the detection filter is f(t) = p∗(T − t) and is thecomplex conjugate of the time-reversed copy of the pulse p(t).The term matched filter refers to the fact that the optimal filter ismatched to the received pulse shape.The sample value is equal to the energy Ep in the pulse p(t).Setting f(t) = p(T − t), (3) becomes

r2

σ2 =

∫∞−∞ p2(t)dtN0/2 =

2EpN0

(5)

where

Ep = r2 =

∫ ∞−∞

p2(t)dt (6)

is the energy in the received pulse p(t), and

σ2 = N0/2, (7)

is the variance of the noise in the sample.Wireless Embedded Systems - Communication Systems Lab - Fall 2017 Lecture 4A 4

Lecture 4A



Bit ErrorRate

Eb/N0Energy Eb

Noise DensityN0



Matched Filters and Nyquist Pulses

For antipodal modulation (BPSK), r1 = −r0 =√Eb where Eb = Ep is

the average energy per bit.Using the minimum square distance d2

min = d210 = (r1 − r0)

2 = 4Eb andσ2 = N0/2, the sample signal-to-noise ratio ζ for a matched filter is

ζ =2d2

10N0


Lecture 4A



Bit ErrorRate

Eb/N0Energy Eb

Noise DensityN0



Waveforms with Raised Cosine Spectra and Matched Filtering

A matched filter whose output p(t)~ p(−t) is a Nyquist pulse has thespecial property that, for white noise at the input, the output Nyquistsamples are uncorrelated.

For best error performance, detection filter is matched to input waveform

To produce no interference waveform after matched filter should beNquist pulse - raised cosine

Both requirements achieved1 TX filter is

√raised cosine

2 RX matched filter is√raised cosine

RX filter is then matched to TX waveform (assuming channel does notalter waveform)

Waveform that is sampled is TX×RX = raised cosine so no interference


Lecture 4A



Bit ErrorRate

Eb/N0Energy Eb

Noise DensityN0



Matched Filter for Square-Wave

Matched filter for square-wave is integrate and dump receiver.

Input signal

Integrated output+noise Threshold

Comparator

Recoveredsymbols

><

Clockrecovery

Reset for next symbol

Sample Time

Waveform integrated over symbol period and sampled at end of period T

Integrator resets for next period.


Lecture 4A



Bit ErrorRate

Eb/N0Energy Eb

Noise DensityN0



31 bit PRBS 8 samples/bit

Waveform before matched filter

Waveform after matched filter (integrate and dump)


Lecture 4A



Bit ErrorRate

Eb/N0Energy Eb

Noise DensityN0



Matched Filter w/noiseWaveform w/noise

Waveform w/noise after matched filter


Lecture 4A



Bit ErrorRate

Eb/N0Energy Eb

Noise DensityN0



Statistical Distributions

Probability Functions

For perfect un-filtered eye-patterns, the difference between the meanvalues s1(T ) and s0(T ) does not depend on where the waveform issampled as long as it is not a transition.


Lecture 4A



Bit ErrorRate

Eb/N0Energy Eb

Noise DensityN0



Gaussian Functions

A gaussian random variable has a gaussian probability distributiondefined by

fx(x).=

1√2πσ

e−(x−〈x〉)2/2σ2

.

with a mean of 〈x〉 and a variance σ2.The probability that a unit-variance gaussian random variable exceeds avalue z, P{x > z}, expressed using the complementary error functiondenoted by erfc

1√2π

∫ ∞z

e−x2/2dx =

12erfc

(z√2

), (8)

where erfc(z) = 1−erf(z) with erf(z) being the error function that isdefined as

erf(z) =2√π

∫ z

0e−s

2ds.

This probability is often expressed using the equivalent function

Q(x).= 1

2 erfc(x/√

2)


Lecture 4A



Bit ErrorRate

Eb/N0Energy Eb

Noise DensityN0



Bit Error Rate

The bit error rate can be determined from s1(T ), s0(T ) and σ

pe =12erfc

[r1 − r02√

2σ

]= pe =

12erfc

[√(r1 − r0)2

8σ2

]= Q(ζ)

Note we have slightly changed definition of ζ from Lab 2 by including afactor of 2.

2 4 6 8 10x

�20

�15

�10

�5

0Q- function (Q(x)

Q(x

)


Lecture 4A



Bit ErrorRate

Eb/N0Energy Eb

Noise DensityN0



Matched Filters and Nyquist Pulses

For antipodal modulation (BPSK), r1 = −r0 =√Eb where Eb = Ep is

the average energy per bit.Using the minimum square distance d2

min = d210 = (r1 − r0)

2 = 4Eb andσ2 = N0/2, the sample signal-to-noise ratio ζ for a matched filter is

ζ2 =d2

104σ2 =

2EbN0

Therefore, the probability of a detection error is

pe =12erfc

[√(r1 − r0)2

8σ2

]=

12erfc

(√d2

104N0

)=

12erfc

(√Eb/N0

).

or in terms of the Q function

pe = Q(ζ) = Q

(√2EbN0

).


Lecture 4A



Bit ErrorRate

Eb/N0Energy Eb

Noise DensityN0



Summary of Assumptions

1 System is AWGN (additive noise)2 Matched filter is used3 If system is bandlimited then waveform before sampling and after

matched filter is raised cosine (or other waveform that has no ISI)4 Sampling is at optimal time

If any of these assumptions is invalid the form of the probability of errorwill not be of the form of

pe = Q

(√2EbN0

)=

12erfc

(√EbN0

)where Q(x)

.= 1

2 erfc(x/√

2)


Lecture 4A



Bit ErrorRate

Eb/N0Energy Eb

Noise DensityN0



Determination of Energy

VI used in lab estimates energy in a bit using

Eb =

L∑i

s[i]2

where s[i] is the sample and L is the number of samples per symbol(which is a bit for binary system)

Assumes all the energy within the bit - not true for raised cosinewaveforms

�3T �2T �T 0 3TTime

0

0.5

1

Res

pons

e

0

Frequency

0

T

Res

pons

e

Β�1Β�1/2Β�0

T 2T12T

12T�1

T�1 T

For β = 0.5 (value used in lab) about half energy outside interval. Needto use correction factor C = 2 so that

Eb = CEb = 2Eb

Energy correct for square-wave waveform C = 1.Wireless Embedded Systems - Communication Systems Lab - Fall 2017 Lecture 4A 15

Lecture 4A



Bit ErrorRate

Eb/N0Energy Eb

Noise DensityN0



Determination of N0

Eb/N0 = EBN0 is the desired value

Eb = CEb is energy per bit with correction factor due to finitesummation.

Required N0 is then

N0 =CEbEBN0

Power spectral density is related to σ of Gaussian random numbergenerator via

N0(W/Hz) = σ2

(fs/2)WHz

where fs is sampling frequency. (Derived and tested in Lab 2.)

If fbit is the symbol rate (bits/s) and L is the number of samples/bit,then

fs = fbitL

Solve for required value of σ to get correct value of N0 in terms of bitrate and samples per bit

σ2 =fbitLN0

2 =fbitLCEb2(Eb/N0)


Lecture 4A



Bit ErrorRate

Eb/N0Energy Eb

Noise DensityN0



Single sample detection

Decision statistic is a single sample of the square-wave eye

Probability FunctionsEye

BER isBER = Q

[r1 − r0

2σ

]= Q

[s

σ

]for antipodal signaling when r1 = −r0 = s

(s/σ)2 is single sample SNR


Lecture 4A



Bit ErrorRate

Eb/N0Energy Eb

Noise DensityN0



Eb and N0 for single sample

Single sample s taken over a time ∆t = 1/fs.

Energy in bit is thenEb = s2∆t = s2/fs

Noise density is (Lab 2)

N0 =2σ2

fs

BER is then

BER = Q

[√2EbN0

]= Q

[√2s2/fs2σ2/fs

]= Q

[s

σ

]as before

“Looks” like matched filter because single sample is constant over ∆t iseffectively an integrator over that single sample.


Lecture 4A



Bit ErrorRate

Eb/N0Energy Eb

Noise DensityN0



Single sample to L samples per bit

Noise spectral density decreases as the number of samples per bit L forfixed σ2

σ2 =fbitLN0

2

Therefore if σ2 in noise generator is fixed, then measured Eb/N0for Lsamples is (

EbN0

)L samples/bit

= L(EbN0

)1 sample/bit

This is a huge difference - evaluted this difference in Prelab 4 question.


Lecture 4A



Bit ErrorRate

Eb/N0Energy Eb

Noise DensityN0



BER Curves

0 2 4 6 8 10

�5

�4

�3

�2

�1

0

(8 samples/bit)

Log

(BER

)One sample/bit

Clock offset of 1 sample

8 samplesno clock offset

~0.25 dBpower

penalty @BER=10-4


Lecture 4A



Bit ErrorRate

Eb/N0Energy Eb

Noise DensityN0



Lab Example

If interpolation is not used, then even the best sampling point is notoptimal

Sample can be at most (∆N/N)Eb off from optimal sampling point.

Lab example: N = 8. No interpolation implies maximum reduction in Ebof

∆Eb =Eb8

or a worst case energy of 78Eb with respect to Eb with no error.

Let Pe = 10−4. The Eb/N0 without an timing error is 6.91. With atiming error, the maximum reduction is

Pe = Q

√2(6.91)× (7/8)︸︷︷︸clock error

= 0.5Erfc(

√6.91× (7/8) = 2.5× 10−4

which is over 2×larger than the Pe without a timing error.


Lecture 4A



Bit ErrorRate

Eb/N0Energy Eb

Noise DensityN0



Average Error Rate

If we assume the timing error is uniform over a range 0 < t < 1/N , thenthe pdf is pt(t) = N and the average error rate is the conditional errorPe|t integrated over the pdf of t or

Pe =N

2

∫ 1/N

0Erfc

[√EbN0

(1− t)]dt

Lab example: N = 8 with Eb/N0 = 6.91

Pe = 4∫ 1/8

0Erfc

[√6.91 (1− t)

]dt

= 1.65× 10−4

Therefore in lab, we expect to see slightly higher Pe because of this effectwith the average increase of 65% and a maximum increase > 200%.


Lecture 4A



Bit ErrorRate

Eb/N0Energy Eb

Noise DensityN0



Threshold Errors

No Threshold Error Threshold Error

One of the terms for the error rate will dominate. In this case it is p0|1


lecture 4a - papenlab1.ucsd.edupapenlab1.ucsd.edu/~coursework/wes/slides/lec4a-matchedfilter.pdf ·...

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