lecture’6 ’mul+carrier’modulaon’and’ofdmweber.itn.liu.se/~vanan11/tnk080/lecture6.pdf ·...

Lecture 6 Mul+carrier Modula+on and OFDM

Prof. Chun-Hung Liu

Dept. of Electrical and Computer Engineering

National Chiao Tung University

Fall 2014

Wireless Communications

14/10/27 Lecture 6: Mul+carrier Modula+on 2

Outline

•  Multicarrier Modulation (Chapter 12 in Goldsmith’s Book) •  Data Transmission using Multiple Carriers •  Multicarrier Modulation with Overlapping Subchannels •  Mitigation of Subcarrier Fading •  Discrete Implementation of Multicarrier •  Challenges in Multicarrier Systems


Introduction •  The basic idea of multicarrier modulation is to divide the transmitted

bitstream into many different substreams and send these over many different subchannels

•  The data rate on each of the subchannels is much less than the total data rate, and the corresponding subchannel bandwidth is much less than the total system bandwidth.

•  The number of substreams is chosen to insure that each subchannel has a bandwidth less than the coherence bandwidth of the channel, so the subchannels experience relatively flat fading. (Why do we need flat fading?)

•  Multicarrier modulation is efficiently implemented digitally. In this discrete implementation, called orthogonal frequency division multiplexing (OFDM), the ISI can be completely eliminated through the use of a cyclic prefix.

•  There are some impairments of multicarrier modulation, such as frequency offset and peak-to-average ratio, etc..


Data Transmission Using Multiple Carriers •  The simplest form of multicarrier modulation divides the data stream into

multiple substreams to be transmitted over different orthogonal subchannels centered at different subcarrier frequencies.

•  The number of substreams is chosen to make the symbol time on each substream much greater than the delay spread of the channel or, equivalently, to make the substream bandwidth less than the channel coherence bandwidth. This insures that the substreams will not experience significant ISI.

•  Consider a linearly-modulated system with data rate R and passband bandwidth B. The coherence bandwidth for the channel is assumed to be so the signal experiences frequency-selective fading.

•  The basic premise of multicarrier modulation is to break this wideband system into N linearly-modulated subsystems in parallel, each with subchannel bandwidth

•  For N sufficiently large, the subchannel bandwidth , which insures relatively flat fading on each subchannel.


Data Transmission Using Multiple Carriers •  Figure 12.1 illustrates a multicarrier transmitter. The bit stream is

divided into N substreams via a serial-to-parallel converter.


Data Transmission Using Multiple Carriers •  The nth substream is linearly-modulated (typically via QAM or PSK) relative to the

subcarrier frequency and occupies passband bandwidth . •  If we assume raised cosine pulses for we get a symbol time for

each substream, where β is the rolloff factor of the pulse shape. •  The modulated signals associated with all the subchannels are summed together to

form the transmitted signal, given as

•  For nonoverlapping subchannels, we set

•  The substreams then occupy orthogonal subchannels with passband bandwidth , yielding a total passband bandwidth and data rate R .

TN = (1 + �)BN

(12.1)


•  The receiver for this multicarrier modulation is shown in Figure 12.2.

Data Transmission Using Multiple Carriers


•  Although this simple type of multicarrier modulation is easy to understand, it has several significant shortcomings. •  In a realistic implementation, subchannels will occupy a larger

bandwidth than under ideal raised cosine pulse shaping since the pulse shape must be time-limited.

•  Let denote the additional bandwidth required due to time-limiting of these pulse shapes.

•  So the total required bandwidth for nonoverlapping subchannels is

•  Thus, this form of multicarrier modulation can be spectrally inefficient. •  Additionally, near-ideal (and hence, expensive) low pass filters will be

required to maintain the orthogonality of the subcarriers at the receiver. •  This scheme requires N independent modulators and demodulators,

which entails significant expense, size, and power consumption.

Data Transmission Using Multiple Carriers

(12.2)


Multicarrier Modulation with Overlapping Subchannels

•  We can improve on the spectral efficiency of multicarrier modulation by overlapping the subchannels.

•  The subcarriers must still be orthogonal so that they can be separated out by the demodulator in the receiver.

•  The subcarriers form a set of (approximately) orthogonal basis functions on the interval for any set of subcarrier phase offsets since



•  It is easily shown that no set of subcarriers with a smaller frequency separation forms an orthogonal set on for arbitrary subcarrier phase offsets.

•  This implies that the minimum frequency separation required for subcarriers to remain orthogonal over the symbol interval is .

•  Consider a multicarrier system where each subchannel is modulated using raised cosine pulse shapes with rolloff factor β.

•  The passband bandwidth of each subchannel is then •  The ith subcarrier frequency is set to for

some , so the subcarriers are separated by . •  However, the passband bandwidth of each subchannel is

for β > 0, so the subchannels overlap. •  Excess bandwidth due to time windowing will increase the subcarrier

bandwidth by an additional .



•  The total system bandwidth with overlapping subchannels is given by

•  Thus, with N large, the impact of β and on the total system bandwidth is negligible, in contrast to the required bandwidth when the subchannels do not overlap.

(for large N) (12.4)



•  In particular, overlapping subchannels are demodulated with the receiver structure shown in Figure 12.4.


Mitigation of Subcarrier Fading •  The advantage of multicarrier modulation is that each subchannel is relatively

narrowband, which mitigates the effect of delay spread. •  However, each subchannel experiences flat-fading, which can cause large BERs

on some of the subchannels. •  In particular, if the transmit power on subcarrier i is , and the fading on that

subcarrier is , then the received SNR is , where is the bandwidth of each subchannel.

•  If is small then the received SNR on the ith subchannel is quite low, which can lead to a high BER on that subchannel.

•  Moreover, in wireless channels the ’s will vary over time according to a given fading distribution, resulting in the same performance degradation associated with flat fading for single carrier systems.

•  Since flat fading can seriously degrade performance in each subchannel, it is important to compensate for flat fading in the subchannels.

•  There are several techniques for doing this, including coding with interleaving over time and frequency, frequency equalization, precoding, and adaptive loading, etc..

Pi↵i

↵i

�i = ↵2iPi/(N0BN ) BN

↵i


Interleaving over Time and Frequency •  The basic idea in coding with interleaving over time and frequency is to encode

data bits into codewords, interleave the resulting coded bits over both time and frequency.

•  Then transmit the coded bits over different subchannels such that the coded bits within a given codeword all experience independent fading.

•  If most of the subchannels have a high SNR, the codeword will have most coded bits received correctly, and the errors associated with the few bad subchannels can be corrected.

•  Coding across subchannels basically exploits the frequency diversity inherent to a multicarrier system to correct for errors.

•  This technique only works well if there is sufficient frequency diversity across the total system bandwidth.

•  If the coherence bandwidth of the channel is large, then the fading across subchannels will be highly correlated, which will significantly reduce the effect of coding.

•  Most coding for OFDM assumes channel information in the decoder. Channel estimates are typically obtained by a two dimensional pilot symbol transmission over both time and frequency.


Frequency Equalization and Precoding •  In frequency equalization the flat fading on the ith subchannel is basically

inverted in the receiver. •  Specifically, the received signal is multiplied by , which gives a resultant

signal power . •  While this removes the impact of flat fading on the signal, it enhances the

noise. Specifically, the incoming noise signal is also multiplied by , so the noise power becomes and the resultant SNR on the ith subchannel after frequency equalization is the same as before equalization.

•  Therefore, frequency equalization does not really change the performance degradation associated with subcarrier flat fading.

•  Precoding uses the same idea as frequency equalization, except that the fading is inverted at the transmitter instead of the receiver.

•  This technique requires that the transmitter have knowledge of the subchannel flat fading gains , which must be obtained through estimation.

•  In this case, if the desired received signal power in the ith subchannel is , and the channel introduces a flat-fading gain in the ith subchannel, then under precoding the power transmitted in the ith subchannel is . The subchannel signal is corrupted by flat-fading with gain , so the received signal power is , as desired.

1/↵i

↵2iPi/↵

2i = Pi

1/↵i

N0BN/↵2i

↵i, i = 10, . . . , N � 1

Pi

↵i

Pi/↵2i

Pi↵2i /↵

2i = Pi

↵i

↵i


Adaptive Loading •  Adaptive loading is based on the adaptive modulation techniques discussed in

Chapter 9. •  It is commonly used on slowly changing channels like digital subscriber lines,

where channel estimates at the transmitter can be obtained fairly easily. •  The basic idea is to vary the data rate and power assigned to each subchannel

relative to that subchannel gain. •  In adaptive loading power and rate on each subchannel is adapted to maximize

the total rate of the system using adaptive modulation such as variable-rate variable-power MQAM.

•  Consider the capacity of the multicarrier system with N independent subchannels of bandwidth and subchannel gain . Assuming a total power constraint P, this capacity is given by:

BN {↵i, i = 0, . . . , N � 1}

(12.7)

•  The power allocation that maximizes this expression is a water-filling over frequency given by Equation (4.24):

Pi


Adaptive Loading (12.8)

•  For some cutoff value , where . The cutoff value is obtained by substituting the power adaptation formula into the power constraint. The capacity then becomes

�0 �i = ↵2iP/(N0BN )

(12.9)

•  Applying the variable-rate variable-power MQAM modulation scheme described in Chapter 9 to the subchannels, the total data rate is given by

(12.10)

where for is the desired target BER in each subchannel. K = �1.5/ ln(5Pb) Pb

•  Optimizing this expression relative to the ’s yields the optimal power allocation Pi

where is a cutoff fade depth dictated by the power constraint P and K. �K


Discrete Implementation of Multicarrier •  Although multicarrier modulation was invented in the 1950’s, its

requirement for separate modulators and demodulators on each subchannel was far too complex for most system implementations at the time.

•  The development of simple and cheap implementations of the discrete Fourier transform (DFT) and the inverse DFT (IDFT) twenty years later, combined with the realization that multicarrier modulation can be implemented with these algorithms, ignited its widespread use.

•  (Review) The DFT and its Properties:

Let x[n], 0 ≤ n ≤ N-1, denote a discrete time sequence. The N-point DFT of x[n] is defined as

The sequence x[n] can be recovered from its DFT using the IDFT:

(12.13)

(12.14)


Discrete Implementation of Multicarrier •  The DFT and its inverse are typically performed in hardware using the fast

Fourier transform (FFT) and inverse FFT (IFFT). •  When an input data stream x[n] is sent through a linear time-invariant

discrete-time channel h[n], the output y[n] is

The N-point circular convolution of x[n] and h[n] is defined as

where denotes [n-k] modulo N. In other words, is a periodic version of x[n-k] with period N.

•  From the definition of the DFT, circular convolution in time leads to multiplication in frequency:

(12.15)

(12.16)

(12.17)


DFT and Cyclic Prefix •  If the channel and input are circularly convoluted then if h[n] is known at

the receiver, the original data sequence x[n] can be recovered by taking the IDFT of Y [i]/H[i], 0 ≤ i ≤ N − 1.

•  Unfortunately, the channel output is NOT a circular convolution but a linear convolution.

•  However, the linear convolution between the channel input and impulse response can be turned into a circular convolution by adding a special prefix to the input called a cyclic prefix.

•  The Cyclic Prefix: •  Consider a channel input sequence x[n]=x[0], . . . , x[N-1] of length N and a

discrete-time channel with finite impulse response (FIR) of length , where is the channel delay spread and the sampling time associated with the discrete time sequence. •  The cyclic prefix for x[n] is defined as {x[N-µ], . . . , x[N-1]}: it consists of

the last µ values of the x[n] sequence.


Discrete Implementation of Multicarrier •  For each input sequence of length N, these last µ samples are appended to

the beginning of the sequence. This yields a new sequence

•  Note that with this definition, which implies that

•  Suppose is input to a discrete-time channel with impulse response h[n]. The channel output y[n], 0 ≤ n ≤ N-1 is then


Discrete Implementation of Multicarrier

•  Thus, by appending a cyclic prefix to the channel input, the linear convolution associated with the channel impulse response y[n] for 0≤ n≤ N-1 becomes a circular convolution.

•  Taking the DFT of the channel output in the absence of noise then yields

and the input sequence x[n], 0 ≤ n ≤ N-1, can be recovered from the channel output y[n], 0 ≤ n ≤ N-1, for known h[n] by


Discrete Implementation of Multicarrier

•  Note that y[n], -µ ≤ n ≤ N-1, has length N+µ, yet the first µ samples y[-µ], . . . , y[-1] are not needed to recover x[n], 0 ≤ n ≤ N-1, due to the redundancy associated with the cyclic prefix.

•  Moreover, if we assume that the input x[n] is divided into data blocks of size N with a cyclic prefix appended to each block to form , then the first µ samples of in a given block are corrupted by ISI associated with the last µ samples of x[n] in the prior block, as illustrated in Figure 12.6.


Discrete Implementation of Multicarrier •  The cyclic prefix serves to eliminate ISI between the data blocks since the

first µ samples of the channel output affected by this ISI can be discarded without any loss relative to the original information sequence.

•  In continuous time this is equivalent to using a guard band of duration (the channel delay spread) after every block of N symbols of duration to eliminate the ISI between these data blocks.

•  The benefits of adding a cyclic prefix come at a cost. Since µ symbols are added to the input data blocks, there is an overhead of µ/N, resulting in a data rate reduction of N/(µ+N).

•  The transmit power associated with sending the cyclic prefix is also wasted since this prefix consists of redundant data.

•  The above analysis motivates the design of OFDM. In OFDM the input data is divided into blocks of size N referred to as an OFDM symbol.

•  A cyclic prefix is added to each OFDM symbol to induce circular convolution of the input and channel impulse response.


Discrete Implementation of Multicarrier Example 12.4: Consider an OFDM system with total passband bandwidth B = 1 MHz assuming . A single carrier system would have symbol time . The channel has a maximum delay spread of µsec, so with µsec, so there would clearly be severe ISI. Assume an OFDM system with MQAM modulation applied to each subchannel. To keep the overhead small, the OFDM system uses N=128 subcarriers to mitigate ISI. So µsec. The length of the cyclic prefix is set to to insure no ISI between OFDM symbols. For these parameters, find the subchannel bandwidth, the total transmission time associated with each OFDM symbol, the overhead of the cyclic prefix, and the data rate of the system assuming M = 16.


Orthogonal Frequency Division Multiplexing (OFDM) •  The OFDM implementation of multicarrier modulation is shown in

Figure 12.7.


Orthogonal Frequency Division Multiplexing (OFDM)



•  The IFFT yields the OFDM symbol consisting of the sequence x[n] = x[0], . . . , x[N-1] of length N, where

•  This sequence corresponds to samples of the multicarrier signal: i.e. the multicarrier signal consists of linearly modulated subchannels, and the right hand side corresponds to samples of a sum of QAM symbols X[i] each modulated by carrier frequency .

•  The transmitted signal is filtered by the channel impulse response h(t) and corrupted by additive noise, so that the received signal is

•  This signal is downconverted to baseband and filtered to remove the high frequency components.


Orthogonal Frequency Division Multiplexing (OFDM) •  The A/D converter samples the resulting signal to obtain −µ ≤ n ≤ N-1. The prefix of y[n] consisting of the first µ samples is then removed.

•  This results in N time samples whose DFT in the absence of noise is

•  The OFDM system effectively decomposes the wideband channel into a set of narrowband orthogonal subchannels with a different QAM symbol sent over each subchannel.

•  An alternative to using the cyclic prefix is to use a prefix consisting of all zero symbols. In this case the OFDM symbol consisting of x[n], 0 ≤ n ≤ N-1 is proceeded by µ null samples, as illustrated in Figure 12.8.

•  At the receiver the “tail” of the ISI associated with the end of a given OFDM symbol is added back in to the beginning of the symbol, which recreates the effect of a cyclic prefix, so the rest of the OFDM system functions as usual.



•  This zero prefix reduces the transmit power relative to a cyclic prefix by , since the prefix does not require any transmit power.

•  However, the noise from the received tail is added back into the beginning of the symbol, which increases the noise power by . Thus, the difference in SNR is not significant for the two prefixes.


Challenges in Multicarrier Systems •  Peak to Average Power Ratio:

•  A low PAR allows the transmit power amplifier to operate efficiently, whereas a high PAR forces the transmit power amplifier to have a large backoff in order to ensure linear amplification of the signal. This is demonstrated in Figure 12.11


Challenges in Multicarrier Systems •  The PAR of a continuous-time signal is given by

and for a discrete-time signal it is given by

•  Any constant amplitude signal, e.g. a square wave, has PAR=0 dB. A sine wave has PAR=3 dB since and

so PAR=1/0.5=2.


Challenges in Multicarrier Systems •  Frequency and Timing Offset: OFDM modulation encodes the data

symbols onto orthogonal subchannels, where orthogonality is assured by the subcarrier separation .

•  The subchannels may overlap in the frequency domain, as shown in Figure 12.12 for a rectangular pulse shape in time.

•  This is generally caused by mismatched oscillators, Doppler frequency shifts, or timing synchronization errors.

•  In practice, the frequency separation of the subcarriers is imperfect: so Δf is not exactly equal to . So intercarrier interference (ICI) could happen.


Challenges in Multicarrier Systems

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