lecture 05: wideband communication -...

I-Hsiang Wang Principle of Communications Lecture 05

Lecture 05: Wideband Communication

Outline

• Frequency selective channel and the equivalent discrete-time baseband channel model• Inter-symbol interference (ISI) and the optimal receiver• Linear equalization: matched filter, zero-forcing, and MMSE• Orthogonal frequency division multiplex (OFDM)

In previous lectures, the channel model for the physical noisy channel only capture the additive noiseeffect: the output signal of the channel Y (t) is the input signal x(t) plus white Gaussian noise Z(t):

Y (t) = hx(t) + Z(t), (1)

where h stands for the attenuation in the channel. Neglecting the noise, the input signal in the frequencydomain always experiences a flat channel: no matter what the operational frequency band is, the channelin the frequency domain remains the same. The reasoning behind the model is that the communicationhappens within a narrow band, so it is reasonable to assume that the channel attenuation is the same atall frequencies. After introducing the analog-digital interface in Lecture 02, the focus is how to combatthe additive noise. We developed the statistical framework for designing the optimal decision makingalgorithm at the receiver in Lecture 03, and further introduced the technique of error correction coding atthe transmitter in Lecture 04. As a result, we have shown that reliable communication is achievable in suchchannels: as long as the data rate is strictly smaller than the channel capacity, as the transmission timegrows, it is possible to make the overall probability of error arbitrarily small.

In this lecture, we consider wideband communications over wireline channels, and the assumption offlat channel is no longer a good fit to the actual channel. Instead, we model the channel as a linear timeinvariant (LTI) filter:

Y (t) = (h ∗ x)(t) + Z(t), (2)

where h(τ) denotes the impulse response of the channel, and Z(t) remains the white Gaussian noise processwith power spectral density SZ(f) = N0/2. Since the channel impulse response is no longer a Dirac deltafunction, its frequency response is no longer flat. This is often called the frequency selective channel, sincethe frequency response is different at different frequencies. We will derive the equivalent complex basebanddiscrete-time channel model, from the input of the pulse shaper at the transmitter, all the way to the outputof the demodulator at the receiver. It turns out the impact of the frequency selectivity of the passbandcontinuous time channel will also introduce frequency selectivity into the equivalent complex basebanddiscrete-time channel. As a result, although the pulse shaping filter and the corresponding receiver filter isdesigned to satisfy the Nyquist criterion so that no inter-symbol interference (ISI) arises when h(τ) = δ(τ),ISI will appear when there is frequency selectivity. ISI will make each of the demodulated symbols becomea linear combination of input symbols from different time indices, and hence introduce memory. Suchdependency on input symbols makes the symbol-by-symbol detection architecture developed in Lecture 03strictly suboptimal, and the impact of ISI becomes more significant in the high SNR regime. See Figure 1for illustration.

The main focus of this lecture is how to mitigate the impact of ISI and perform better detection in thedigital world. We will first introduce receiver-centric techniques for ISI mitigation, including the optimalmaximum likelihood sequence detection (MLSD) implemented using Viterbi algorithm, and the family oflinear equalizations including the matched filter (MF) equalizer, the zero-forcing (ZF) equalizer, and the

Fall 2017 National Taiwan University


ECC Encoder

Symbol Mapper

Pulse Shaper

Filter + Sampler + Detection

Symbol Demapper

ECC Decoder

codedbits

discrete sequence

Binary Interface

Channel Coding

Information bits

Up Converter

Down Converter

baseband waveform Noisy

Channel

passband waveform

x(t)

Y (t)

LTI filter + noise

ISI happens and deteriorate detection!

Figure 1: Frequency selective channel and the inter-symbol interference

minimum mean squared error (MMSE) equalizer. Instead of minimizing the probability of error, these linearschemes try to optimize certain objective functions related to the energy of signal, interference, and noise.Finally, we introduce a solution where both the transmitter and the receiver are involved, called orthogonalfrequency division multiplexing (OFDM), which convert the ISI channel in time domain to parallel channelsin the frequency domain.

1 Frequency selective channel

The main channel model considered in this lecture is given by Eq. (2), where the key element is the LTIfilter h(τ) (the impulse response). There are several reasons to use a linear time invariant filter to model awireline channel:

• Frequency response is no longer flat• Channel is rather stationary within the relevant time of communication• Transmission lines can be approximated as a linear system

h(⌧)

⌧time dispersion (delay spread)Td

=) Y (t) =

Z Td

0h(⌧)x(t� ⌧) d⌧ + Z(t)

Figure 2: Impulse response of a frequency selective channel

As a result, there are two key features of the impulse response h(τ):



(a) Causal: naturally, the output of the channel at time t should not depend on the future input signalsafter time t. Hence, the filter should be causal, and its impulse response h(τ) = 0 for all τ < 0.

(b) Dispersive: the wires or cables should not be able to “keep” the signal forever. In other words, theinput signal cannot “stay” in the channel for too long, and hence there exists a finite Td such thath(τ) = 0 for all τ > Td. This finite Td is called the time dispersion (or delay spread) of the channel.Typically Td depends only on the material of wires used in the wireline communication. For example,for telephone and Ethernet wires, Td is roughly tens of microseconds (µs).

In short, most energy of the impulse response of the channel should be contained in an interval [0, Td]. SeeFigure 2 below for illustration.

1.1 Equivalent complex baseband continuous-time channel

First let us derive the equivalent complex baseband continuous-time channel neglecting the noise, where theinput is

xb(t) = x(I)b (t) + jx(Q)

b (t) ∈ C, (3)

the complex baseband representation of the input signal, and the output is

yb(t) = y(I)b (t) + jy(Q)

b (t) ∈ C. (4)

x(I)b (t) and x

(Q)b (t) are the baseband modulated waveforms from the pulse shaping.

x(I)b (t) ≜

∑m

u(I)m p(t−mT ), x(Q)b (t) ≜

∑m

u(Q)m p(t−mT ). (5)

Hence, xb(t) ≜∑

m um p(t−mT ) with um = u(I)m + ju(Q)

m . y(I)b (t) and y

(Q)b (t) are the down-converted signals

from the cos (in-phase) branch and the sin (quadrature) branch respectively.Let us now go through the entire process from the up-conversion at the transmitter to the down-

conversion at the receiver:

(a) Up conversion: let xp(t) ≜ xb(t)√2 exp(j2πfct). Then,

x(t) = Re {xp(t)} =1

2

{xp(t) + x∗p(t)

}. (6)

(b) Frequency selective channel:

y(t) = (h ∗ x)(t) = 1

2

{(h ∗ xp)(t) + (h ∗ x∗p)(t)

}=

1

2

{(h ∗ xp)(t) + (h∗ ∗ x∗p)(t)

}(7)

= Re {(h ∗ xp)(t)} . (8)

Since

(h ∗ xp)(t) =∫ Td

0h(τ)xb(t− τ)

√2 exp(j2πfc(t− τ))dτ (9)

=√2ej2πfct

∫ Td

0h(τ)e−j2πfcτxb(t− τ)dτ (10)

=√2ej2πfct(hb ∗ xb)(t), where hb(τ) ≜ h(τ)e−j2πfcτ , (11)

we have shown thaty(t) = Re

{(hb ∗ xb)(t)

√2 exp(j2πfct)

}. (12)



(c) Down conversion: from Eqn. (12), we can directly conclude that after down conversion,

yb(t) = (hb ∗ xb)(t). (13)

In other words, the complex-baseband filter hb(τ) ≜ h(τ)e−j2πfcτ is just the down-converted versionof the passband filter h(τ).

In summary, the equivalent complex baseband continuous-time channel (neglecting the noise) is givenby Eqn. (13).

1.2 Equivalent complex baseband discrete-time channel

To derive the equivalent complex baseband discrete-time channel, we first recall that the baseband waveformis formed by pulse shaping: xb(t) =

∑m um p(t−mT ). At the receiver, after down conversion, the next step

is to conduct digital demodulation, which gives

Vm = um + Zm, Zmi.i.d.∼ CN (0, N0), ∀m ∈ Z. (14)

As for um, its relationship with the input symbols {um} can be derived as follows: let q(t) denote the receiverPAM filter, then the filtering and the sampling operations yield

um = (yb ∗ q)(mT ) = (xb ∗ hb ∗ q)(mT ) (15)

=∑k

uk

∫ Td

0hb(τ)g(mT − kT − τ)dτ (16)

=∑k

ukhd[m− k] =∑ℓ

hd[ℓ]um−ℓ = (hd ∗ u)m, (17)

where g(t) ≜ (p ∗ q)(t) denotes the overall pulse, and hd[ℓ] ≜ (hb ∗ g)(ℓT ) denotes the equivalent discrete-time filter of the discrete-time baseband channel model. Hence, the equivalent complex baseband discrete-time channel is characterized by

Vm =∑ℓ

hd[ℓ]um−ℓ + Zm, Zmi.i.d.∼ CN (0, N0), ∀m ∈ Z. (18)

We can now see better what is the impact of the frequency selectivity in the passband channel. Recallin Lecture 02, when h(t) = δ(t), the effective discrete-time filter is hd[ℓ] = g(ℓT ) = δℓ = 1 {ℓ = 0} by theideal Nyquist property. As a result, um = um for all m, and there is no ISI at all. On the other hand, whenthere is frequency selectivity in h(τ), even if g(t) is ideal Nyquist with interval T , ISI-free is not guaranteed.

1.3 Effective number of taps

Next, let us take a deeper look into the discrete-time filter {hd[ℓ] | ℓ ∈ Z}. We call the filter coefficients “taps”of the filter. Each tap of the filter is the outcome of the continuous-time baseband filter hb(t) convolves withthe pulse g(t) and gets uniformly sampled with interval T . Since both hb(t) and g(t) have a finite “span”,that is, most energy are constrained within a finite duration of time, hd[ℓ] will be very close to zero when ℓin not in a specific range. In other words, it is fine to approximate the discrete-time filter {hd[ℓ]} as a causalfinite impulse response (FIR) filter, that is,

hd[ℓ] = 0, ∀ ℓ /∈ {0, 1, . . . , L− 1}. (19)

Note that the causality can be assumed without loss of generality since we can always shift the time index.



The number L is the number of taps in this discrete-time LTI filter channel. Since hd is the sampledoutcome of (hb ∗ g)(t), L depends on the support (range of non-zero response) of the convolved (hb ∗ g)(t).Suppose Tp denotes the length of the interval in which most energy in g(t) lies (see Figure 3 for illustration).After convolution, the effective spread becomes roughly Tp + Td. Hence, the number of taps

L ≈ Tp + TdT

= 2WTd +Tp

T, (20)

where Tp

T is roughly a constant depending only on what kind of pulse is chosen.

h(τ)

τ

Td

g(t)

t

Tp

Figure 3: Delay spreads of the frequency selective channel and the pulse

In summary, by identifying hd[ℓ] ≡ hℓ in the rest of this lecture, the equivalent discrete-time complexbaseband channel model to be dealt with is described as follows:

Vm =L−1∑ℓ=0

hℓum−ℓ + Zm, Zmi.i.d.∼ CN (0, N0), ∀m. (21)

Remark (Why ISI matters in wideband communication). From Eqn. (20), we can directly see why ISI ismore critical as the bandwidth W increases. The larger W is, the larger L is, and hence the more sever ISIis. In words, ISI is critical in wideband communication.

2 Inter-symbol interference and the optimal receiver

Let us rewrite Eqn. (21) as follows:

Vm = h0um + (h1um−1 + ...+ hL−1um−L+1) + Zm. (22)

When L = 1, there is no ISI:Vm = h0um + Zm, (23)

and hence symbol-by-symbol detection is optimal because Zmi.i.d.∼ CN (0, N0) for all m. When L becomes

large, the additional ISI termIm ≜ h1um−1 + ...+ hL−1um−L+1 (24)

will degrade the performance of the symbol-wise detector, since the detector for um will treat Im as noise.Treating ISI Im as noise has two major issues. First, the energy of the interference is comparable with

the signal energy. Suppose the symbols are i.i.d. and uniformly distributed over a symmetric constellation



set such as QAM, PSK so that E [Um] = 0. Let the average energy per symbol is Es, that is, Es = E[|Um|2

].

Then, the ISI energy is

E[|Im|2

]=

L−1∑ℓ=1

|hℓ|2E[|Um−ℓ|2

]=

(L−1∑ℓ=1

|hℓ|2)Es. (25)

Hence, the signal-to-interference ratio is

|h0|2Es(∑L−1ℓ=1 |hℓ|2

)Es

=|h0|2(∑L−1ℓ=1 |hℓ|2

) , (26)

does not grow as signal energy grows. Second, notice that ISI comprises past symbols that carry information,and it may subject to huge loss of information if one treats it as pure noise.

Hence, instead of symbol-wise detection, the optimal receiver should detect the entire sequence ofsymbols {um} based on the observed sequence of symbols {Vm}. The optimal detecter is the maximumlikelihood sequence detection (MLSD), which we introduced in Lecture 04 in the context of decodingconvolutional codes.

To be more formal, let the transmitter sends out a length-n sequence of symbols (u1, u2, · · · , un) andthe receiver observes a length-(n+ L− 1) sequence of symbols (V1, V2, . . . , Vn+L−1), where

V1 = h0u1 + Z1

V2 = h0u2 + h1u1 + Z2...

VL = h0uL + h1uL−1 + · · · + hL−1u1 + ZL

VL+1 = h0uL+1 + h1uL + · · · + hL−1u2 + ZL+1...

Vn = h0un + h1un−1 + · · · + hL−1un−L+1 + Zn

Vn+1 = h1un + · · · + hL−1un−L+2 + Zn+1...

Vn+L−1 = hL−1un + Zn+L−1

. (27)

The above can be equivalently represented in the matrix form: V = hu+Z, where

V ≜

V1

V2...

Vn−L+1

, u ≜

u1u2...un

, Z ≜

Z1

Z2...

Zn−L+1

, h ≜

h0 h1 · · · hL−1 0 · · · 00 h0 · · · hL−2 hL−1 · · · 0... . . .0 · · · h0 · · · hL−2 hL−1

⊺

. (28)

In particular, the (n+L−1)×n-matrix h consists of down-shift columns of the FIR filter vector h⊺, whereh = [h0 h1 · · · hL−1]. Since Z1, . . . , Zn+L−1

i.i.d.∼ CN (0, N0), the MLSD rule boils down to the minimumEuclidean distance rule, and is described as follows: (A denotes the constellation set for modulation)

u(V ) = argminu∈An

∥V − hu∥ . (29)

Notice that hu is the linear algegraic representation of the linear convolution between the input sequence{um} and the FIR filter h. Hence, this operation can be equivalently represented as a finite state machineas in the convolutional encoder, where the state and the output at time m are defined as

sm ≜ (um−1, . . . , um−L+1) and om ≜ h0um +L−1∑ℓ=1

hℓ um−ℓ (30)



respectively. In other words, each candidate sequence can be represented as a path on a (n + L − 1)-hoptrellis with |A|L−1 possible states in each layer, and the goal of the MLSD is to find a path on the trellissuch that the aggregate cost

∥V − hu∥2 =n+L−1∑m=1

|Vm − om|2 (31)

is minimized. Note that we use the convention um = 0 for m /∈ {1, . . . , n}. The above MLSD problem canhence be solved optimally using Viterbi algorithm, as described in Lecture 04.Exercise 1. For a two-tap LTI channel h = [h0 h1], suppose the transmitter employs 4-QAM (QPSK).

(a) Define the state and draw the state transition diagram that represents the convolution in the LTIchannel.

(b) Draw the trellis diagram for MLSD when n = 4.

In practice, however, MLSD is used for ISI mitigation only for narrowband communications such asvoice dial-up modem. This is becuase the number of taps L is proportional to the bandwidth 2W , and forwideband communication systems such as Ethernet cables and TV cables, the bandwidth is hundreds ofMHz, while the time dispersion Td is tens of µs. As a result, the number of taps L ≈ 102 − 103. Since thecomputational complexity of Viterbi algorithm is exponential in L, it is infeasible in practice.

We close this section by the following remark.Remark. Since the optimal MLSD is practically infeasible for mitigating ISI in wideband communicationover a frequency selective channel, it hints that one should relax the goal of minimizing the probability oferror (recall the MAP and ML rules are derived to minimize the probability of error) and look for otherheuristics. In the next two sections, we introduce two different paradigms for ISI mitigation. Both of themtry to “convert” the original channel into other kinds of channels, where ISI is either mitigated and treatedas noise, or completely removed subject to some additional costs.

3 Linear equalization

{Vm} Linear Equalizer LTI filter: {g`}

{Wm} symbol-wise detection

{um}

Figure 4: Linear equalization: receiver architecture

In this section, we introduce a family of receiver-centric techniques for ISI mitigation, called linearequalization. The high-level idea is pretty simple: since ISI is caused by the equivalent discrete-time complexbaseband filter {hℓ ∈ C | ℓ = 0, 1, . . . , L − 1}, why not use another discrete-time linear filter {gℓ} tomitigate the ISI at the receiver? See Figure 4 for illustration of the architecture. Such linear filters are alsocalled “linear equalizers” in the communication literature. In the following, we will introduce three kindsof most common linear equalizers: matched filter (MF), zero forcing (ZF), and minimum mean squarederror (MMSE). We shall use the Z transform to describe the linear filters: recall that the Z transform of adiscrete-time sequence {gℓ} is

g(ζ) ≜∑ℓ

gℓζ−ℓ. (32)

Furthermore, the discrete-time Fourier transform can be written in terms of its Z transform: g(f) = g(ej2πf ).Note the subtle notational difference between (·) and (·).



The key design question is: what is the objective when one design the linear filter {gℓ}? In other words,what are we trying to optimize? Since we are doing symbol-wise detection after the linear filtering (whichis by no means optimal), the performance plausibly depends on the signal-to-interference-and-noiseratio (SINR). As we will see, the “best” linear filter that maximizes SINR in the filtered signal {Wm} isthe MMSE equalizer.

The linear equalizers to be introduced are:

• Matched filter (MF):g(MF)(ζ) = h∗(1/ζ∗). (33)

• Zero forcing (ZF):g(ZF)(ζ) = (h(ζ))−1. (34)

• Minimum mean squared error (MMSE):

g(MMSE)(ζ) =Esh

∗(1/ζ∗)

N0 + Esh∗(1/ζ∗)h(ζ). (35)

It is not hard to see that in the low SNR regime, that is, Es ≪ N0, g(MMSE)(ζ) ≈ EsN0

g(MF)(ζ). Meanwhile,in the high SNR regime (Es ≫ N0), g(MMSE)(ζ) ≈ g(ZF)(ζ).

In the following, we first focus on the low SNR and high SNR regimes, show that the optimal linearequalizers that maximizes SINR are MF and ZF respectively. We shall employ a linear algebraic approachand provide the geometric interpretation. Then, we tackle the general case and derive the optimal linearequalizer: MMSE.

3.1 Low SNR regime: matched filter

In this and the next sub-section, we shall employ a linear algebraic approach as in Eqn. (27) and (28). Tobe compatible with the filter perspective, we shall assume that n ≫ L and focus on detecting um for anL≪ m≪ n. Now, Eqn. (27) can be rewritten as follows:

V = um[h]m +∑i =m

ui[h]i +Z, (36)

where [h]m denotes the m-th column vector of the (n+ L− 1)× n-matrix h defined in Eqn. (28), that is,its m-th to (m+L−1)-th entries are h0, h1, . . . , hL−1 while other entries are all zeros. The convolution witha LTI filter can be viewed as inner product with another column vector [g]m:

Wm = ⟨V , [g]m⟩ = [g]HmV . (37)

Concatenating all the column vectors, we arrive at another matrix g whose conjugate1 of the columnswill equivalently describe the LTI equalization filter {gℓ}, just like the matrix h in Eqn. (28) equivalentlydescribing the LTI channel filter {hℓ}.

Note that the energy of ISI is proportional to the average transmit symbol energy Es. Hence, in the lowSNR regime (Es ≪ N0), we can neglect the ISI term. Hence, Eqn. (36) can be rewritten as follows:

V = um[h]m +Z =⇒ Wm = ([g]Hm[h]m)um + Zm, (38)

where Zm ≜ [g]HmZ. The SINR becomes just the SNR in Wm:

SINR =

∣∣[g]Hm[h]m∣∣2

∥[g]m∥2Es

N0=

(|⟨[h]m, [g]m⟩|∥[g]m∥

)2 Es

N0. (39)

1The extra conjugation is because in convolution there is no conjugation involved, unlike inner product.



Note that geometrically, the ratio |⟨[h]m,[g]m⟩|∥[g]m∥ is just the length of the projection of the signal vector [h]m

onto the direction of [g]m. Hence, To maximize SINR, one should choose the projection direction in parallelwith the signal vector [h]m, that is,

[g(MF)]m = [h]m. (40)

As a result, g(MF) = h, and

Wm = h∗0Vm + h∗1Vm+1 + . . .+ h∗L−1Vm+L−1 (41)

=

L−1∑ℓ=0

h∗ℓVm+ℓ =

0∑ℓ=−(L−1)

h∗−ℓVm−ℓ =

0∑ℓ=−(L−1)

g(MF)ℓ Vm−ℓ, (42)

where g(MF)ℓ = h∗−ℓ. Hence, in terms of Z transform, the optimal filter can be characterized as in Eqn. (33).

The name “matched filter” is clear from the geometric picture: the optimal linear equalizer in the lowSNR regime just project the signal onto the signal direction, so that the signal energy is maximized.Since noise dominates interference, the optimal equalizer should focus on maximizing signal energy. Hence,the equalizer’s “direction” is matched with the signal direction.

3.2 High SNR regime: zero forcing

interference subspace

[g( )]m ⌘ [h]m

[g( )]m

[g( )]m

Figure 5: Geometric interpretation of linear equalization

In the high SNR regime, Es ≫ N0, and hence the interference term dominates the noise term. We canaccordingly neglect the noise terms, and rewrite Eqn. (36) as follows:

V = um[h]m +∑i =m

ui[h]i. (43)

Consider the column space spanned by h\m, the submatrix of h after removing the m-th column [h]m:

Im ≜ Column Space of h\m. (44)

This is called the “interference subspace” for symbol um. In other words, it is the subspace spanned by allthe directions of interfering symbols {ui | i = m}. In our setting, since there are n− 1 interfering symbols,the dimension of Im is (n− 1), while the overall dimension of the vector space of V is (n+ L− 1). Hence,



the null space of the (n− 1) interfering directions has dimension L. It is possible to find one direction sothat after projecting onto it, all the interferences are forced to zero, and the outcome becomes a scaledversion of um without any ISI left.

To be more specific, we can choose the zero-forcing vector for symbol um as follows:

[g(ZF)]m = (h†)Hem = h(hHh)−1em, (45)

where h† ≜ (hHh)−1hH denotes the pseudoinverse of the “tall” matrix h, and em denotes the unit vectorwith the m-th entry being 1. It is not hard to verify

⟨[h]i, [g(ZF)]m⟩ = [g(ZF)]Hm[h]i = e⊺mh†[h]i = e⊺m(h†h)ei = e⊺mei =

{1, i = m

0, i = m. (46)

Figure 5 gives a geometric illustration of the matched filter equalizer and the zero forcing equalizer.As for the filter view, it is pretty simple to see that the Z transform of the zero forcing equalizer should

be the inverse of the Z transform of the channel filter, as characterized in Eqn. (34).

3.3 Maximizing SINR: MMSE equalizer

What is the optimal linear equalizer that maximizes SINR? In the following, we will first argue that linearequalization can be viewed as linear estimation of the transmitted symbols {um}. As a result, maximizingSINR is equivalent to minimizing the mean squared error (MSE) of the linear estimator. In other words, theoutput symbols of the linear equalizer (LTI filter), termed {wm}, serve as the estimation of the transmittedsymbols {um}. Then, we derive the optimal LTI filter that renders the minimum MSE, which is termed“MMSE equalizer”.

3.3.1 Maximizing SINR ≡ minimizing MSE

We begin with the formulation of the problem. Here we model the input symbol sequence {Um | m ∈ Z}as a discrete-time random process, and Um

i.i.d.∼ Unif{A} where A denotes the constellation set. The outputrandom process of the frequency selective channel is {Vm | m ∈ Z}, where

Vm = (h ∗ U)m + Zm, Zmi.i.d.∼ CN (0, N0), m ∈ Z. (47)

The linear equalizer is just a LTI filter {gk | k ∈ Z}, and the outcome of the equalizer is {Wm | m ∈ Z},where

Wm = (g ∗ V )m =∑k∈Z

gkVm−k =∑k

L−1∑ℓ=0

gkhℓUm−k−ℓ +∑k

gkZm−k (48)

=

(L−1∑ℓ=0

g−ℓhℓ

)Um + Im + Zm, (49)

where Im and Zm denote the effective ISI and noise terms. We can see that the coefficient of the desiredsymbol, Um, is

∑L−1ℓ=0 g−ℓhℓ, and it does not depend on m. Hence, when finding the best equalizer {gk |

k ∈ Z}, one can assume without loss of generality that the coefficient of the desired symbol at the output isalways 1. As a result,

Wm = Um + Im + Zm = Um +Ξm, (50)



where Ξm ≜ Im + Zm denotes the estimation error of using Wm as the estimate of Um. The signal-to-interference-and-noise ratio is hence

SINRm =E[|Um|2]

E[|Im|2] + E[|Zm|2]=

EsE[|Ξm|2]

. (51)

Therefore, maximizing SINRm is equivalent to minimizing the mean squared error

MSEm ≜ E[|Ξm|2]. (52)

3.3.2 MMSE estimation

The problem of deriving the estimator that minimizes the mean squared error is a very general problem inestimation theory. Here let us give an overview of this general problem. See Figure 6 for illustration of theMMSE estimation problem.

g( )(·) = argming2H

MSE(X, g(Y ))

observation

YEstimator in H

g(·) X = g(Y )

target

X

estimation

MSE(X, X) , E

��X � X��2�

random processes random vectors

{Xn}, {Yn}X,Y

H

LTI filter (FIR/IIR, causal/non-causal) general functions/linear functions

PX,Y

Figure 6: MMSE estimation problem

For the case of finite-horizon observation/estimation, it can be viewed as estimating a random vectorX (target) from a random vector Y (observation). For the case of infinite-horizon observation/estimation,it can be viewed as estimating a random process {Xn} (target) from a random process {Yn} (observation).The estimator is a function g(·) of the observation, and usually the estimator is restricted within certainfunctional classes. For example, in the most general case, the estimator is restricted to be a measurablefunction. In other cases, the estimator might be restricted to be a linear function, a LTI filter, a causal LTIfilter, a FIR causal filter, etc.. Such restrictions vary with the design criteria of the estimation problems.

A particularly well-known result from undergraduate probability and statistic is that, when there is norestriction on the estimator (only need to be measurable), a MMSE estimator is the conditional expectationof X given Y = y:

XMMSE(y) = E [X|Y = y] . (53)If the estimator is restricted to be linear functions, then a linear MMSE estimator is given by the followingformula:

XLMMSE(y) = E [X] + KX,Y K−1Y (y − E [Y ]), (54)

where KX,Y ≜ Cov(X,Y ). To learn more details, please refer to Chapter 10 of [1]2.For the scope of our problem, both the target and the observation are wide-sense stationary (WSS)

random processes. The MMSE estimator is restricted to LTI filters. The optimal filter is called the WienerFilter. In the following, we briefly recap a few useful properties of WSS random processes that are essentialto the derivation of Wiener Filter. Then, we derive the non-causal IIR Wiener filter. The derivation of FIRand causal IIR Wiener filters is beyond the scope of this course, and hence neglected.

2A free draft of this book can be found online.



WSS random process The first and second moments of a discrete-time random process {Xn} are definedas follows:

First Moment µX [n] ≜ E[Xn] (Mean)Second Moment RX [n1, n2] ≜ E[Xn1X

∗n2] (Auto-correlation) (55)

A random process {Xn} is wide sense stationary (WSS) if

Mean µX [n] ≡ µX , ∀n ∈ ZAuto-correlation RX [n+ k, n] ≡ RX [k], ∀n, k ∈ Z (56)

Two random processes {Xn} and {Yn} are jointly WSS if they are both WSS and the cross correlation

RX,Y [n+ k, n] ≡ RX,Y [k], ∀n, k ∈ Z (57)

where RX,Y [n1, n2] ≜ E[Xn1Y∗n2]. For jointly WSS processes {Xn} and {Yn}, the power spectral densities

(PSD) and the cross PSD are defined as the Z transform or the discrete-time Fourier transform (DTFT) ofthe second moments:

RX [k]Z←→ SX(ζ)

RX,Y [k]Z←→ SX,Y (ζ)

(58)

Exercise 2. Show the following properties for two jointly WSS random processes {Xn} and {Yn}.

(a) RX [−k] = R∗X [k] and SX(ζ) = S∗

X(1/ζ∗).(b) RY,X [k] = R∗

X,Y [−k] and SY,X(ζ) = S∗X,Y (1/ζ

∗).Proposition 1 (Filtering of WSS Random Processes): For two jointly WSS random processes {Un}, {Vn}and two LTI filters {hℓ}, {gℓ}, let Xn ≜ (h ∗ U)n and Yn ≜ (g ∗ V )n. Then, {Xn} and {Yn} are also jointlyWSS. Furthermore,

RX,Y [k] =(h ∗ RU,V ∗ g(MF)

)k, SX,Y (ζ) = h(ζ)SU,V (ζ)g

∗(1/ζ∗), (59)

where g(MF)ℓ = g∗−ℓ.

Derivation of the non-causal IIR Wiener filter We are ready to derive the MMSE LTI filter. Considerthe MMSE estimation problem via LTI filtering (depicted in Figure 7) where the target process is {Xn},the observation process is {Yn}, and we assume {Xn} and {Yn} are jointly WSS. First, recall that MSEn ≜E[|Ξn|2

]≡ MSE does not depend on n because the estimation error process {Ξn}, where Ξn = Xn−(g∗Y )n,

is also WSS. The mean squared error can be further manipulated as follows:

MSE = E [Ξn(Xn − (g ∗ Y )n)∗] =

∑k∈Z

g∗kE [Ξn(Xn − Yn−k)∗] (60)

To find the best filter coefficients {gk | k ∈ Z} so that MSE is minimized, one needs to solve the followingoptimization problem:

{g(MMSE)k } = argmin

{gk}MSE. (61)

Note that there are no constraints on the filter coefficients. Also, note that MSE is a convex quadraticfunction of the filter coefficients. To solve such an unconstrained convex optimization problem, it suffices tofind out the stationary points of the quadratic function. For finding the stationary points of a real-valuedfunction of complex-valued variables, we can apply the following theorem taken from [2].Theorem 2: Let f(z, z∗) be a real-valued function of the complex vector z and its conjugate z∗. Then,∇z∗f(z, z∗) lies on the direction of the maximum rate of change of f(z, z∗). As a result, the stationarypoints can be found by solving the equation ∇z∗f(z, z∗) = 0.



Estimation via Linear Filter{Xn} {Yn}

jointly WSS

{gk} ! g(⇣){Xn} = {(g ⇤ Y )n}

⌅n

MSE , E

��Xn � Xn

��2�

also WSS!

Figure 7: Wiener filter: MMSE estimation via LTI filtering

To solve the MMSE problem in (61), we find the stationary point by finding filter coefficients {gk}satisfying the the following:

∀ k ∈ Z, 0 =∂

∂g∗kMSE = −E

[ΞnY

∗n−k

]= E

[(g ∗ Y )nY

∗n−k

]− E

[XnY

∗n−k

](62)

⇐⇒ ∀ k ∈ Z, (g ∗ RY )[k] = RX,Y [k] ⇐⇒ g(ζ)SY (ζ) = SX,Y (ζ). (63)

Hence, the MMSE filter (non-causal IIR Wiener Filter) is given by

g(MMSE)(ζ) =SX,Y (ζ)

SY (ζ). (64)

Orthogonality principle As mentioned in Chapter 10 of [1], one can view the collection of all finite-energy complex random variables as a complex-valued inner-product space, where the inner product betweentwo random variables is defined as

⟨X,Y ⟩ ≜ E[XY ∗]. (65)With the above point of view, one of the key steps in the above derivation, Eqn. (62), worths a second look.Note that Eqn. (62) is equivalent to the following.

∀ k ∈ Z, E[ΞnY

∗n−k

]= 0 ⇐⇒ ⟨Ξn, (g ∗ Y )n⟩ = 0, ∀LTI filter {gk} (66)

In words, the estimation error produced by the MMSE estimator should be orthogonal to all LTI filteredoutcome of {Yn}. If we view the estimators over which we search as a “subspace” of random variables({g(Y ) | g ∈ H}), then geometrically, the MMSE estimator is the projection of the target X onto theestimator subspace. Moreover, the MMSE is just the shortest distance of the target X to the estimatorsubspace. By the Pythagoras Theorem, it is immediate to see that

min MSE + E[∣∣∣XMMSE(Y )− g(Y )

∣∣∣2] = E[|X − g(Y )|2

], ∀ g ∈ H. (67)

See Figure 8 for illustration.

Minimum MSE Using the orthogonal principle, the derivation of the minimum MSE can be slightlysimplified

min MSE = E [ΞnΞ∗n] = E [ΞnX

∗n] = E [XnX

∗n]− E

[(g(MMSE) ∗ Y )nX

∗n

](68)

= RX [0]−∑k

g(MMSE)k RY,X [−k] = RX [0]− (g(MMSE) ∗ RY,X)[0] (69)

=

∫ 12

− 12

(SX(f)− g(MMSE)(f)SY,X(f)

)df =

∫ 12

− 12

(SX(f)− |SXY (f)|2

SY (f)

)df. (70)



estimator subspace

targetX

X (Y )

Ξ

Figure 8: Orthogonality principle

3.3.3 MMSE Equalizer

We are now ready to derive the optimal linear equalizer by applying the Wiener Filter found in (64). Let usidentify {Um} as the target process and {Vm} is the observed process. We need to compute the PSD SV (ζ)and the cross PSD SU,V (ζ). Note that Vm = (h ∗ U)m + Zm, {Zm}, {Um} mutually independent, and thefact that {Vm} is an i.i.d. process with average energy Es. Hence,

SV (ζ) = h(ζ)SU (ζ)h∗(1/ζ∗) + SZ(ζ), SU,V (ζ) = SU (ζ)h

∗(1/ζ∗). (71)

As a result,

g(MMSE)(ζ) =SU,V (ζ)

SV (ζ)=

Esh∗(1/ζ∗)

Esh∗(1/ζ∗)h(ζ) +N0

(72)

The resulting maximum SINR is just Esmin MSE , where

min MSE =

∫ 12

− 12

(SU (f)−

|SU,V (f)|2

SV (f)

)df =

∫ 12

− 12

Es −

∣∣∣h(f)∣∣∣2E2s∣∣∣h(f)∣∣∣2Es +N0

df (73)

= Es

∫ 12

− 12

df|h(f)|2 Es

N0+ 1

. (74)

Hence,

max SINR =

(∫ 12

− 12

(∣∣∣h(f)∣∣∣2 Es

N0+ 1

)−1

df)−1

. (75)

4 OFDM

So far all the introduced ISI mitigation techniques are receiver-centric, that is, the transmitter (Tx) sendsout symbols as if there is no inter-symbol interference, and the receiver (Rx) needs to mitigate ISI on itsown. In general, Tx is also aware of ISI and hence can do some “pre-processing” to mitigate ISI. A completetransmitter-centric approach for ISI mitigation is Tomlinson-Harashima precoding [3, 4]. More generally,Tx and Rx can jointly mitigate ISI, where Tx pre-processes the transmitted signals and Rx post-processes



the received signals, so that the end-to-end equivalent channel becomes ISI-free. It turns out that suchjoint mitigation of ISI is possible as long as the system deliver information in the frequency domain. Inthis section, we introduce orthogonal frequency division multiplexing (OFDM), an elegant approach for ISImitigation. OFDM is widely used in wideband communications nowadays.

4.1 Prelude: free of ISI in the frequency domain

As a prelude to OFDM, the following basic fact in Signals and Systems tells us that we can use In-verse Discrete-Time Fourier Transform (IDTFT) and Discrete-Time Fourier Transform (DTFT) as thepre-processor and post-processor to achieve an end-to-end ISI-free channel in the frequency domain. Itbasically says that complex sinusoids are eigenfunctions to any LTI filter.Theorem 3: A discrete-time complex sinusoid ϕ

(f)n ≜ exp(j2πfn) is an eigenfunction to any LTI filter, that

is, for any impulse response {hℓ}, (ϕ(f) ∗ h

)n= h(f)ϕ(f)

n , (76)

where h(f) ≜∑

ℓ hℓe−j2πfℓ is the DTFT of the impulse response {hℓ}.

Proof By definition, (ϕ(f) ∗ h)n =∑

ℓ hℓ ej2πf(n−ℓ) =

(∑ℓ hℓ e

−j2πfℓ) exp(j2πfn) = h(f)ϕ(f)n .

As a corollary to the above theorem, the DTFT of the convolution of two discrete-time sequences is themultiplication of their DTFT.Corollary 4: Let yn ≜ (x ∗ h)n. Then, y(f) = x(f)h(f).

Proof By definition,

xn =

∫ 1

0x(f)ej2πfn df =

∫ 1

0x(f)ϕ(f)

n df, (77)

and hence

yn = (x ∗ h)n =

∫ 1

0x(f)

(ϕ(f) ∗ h

)n

df =

∫ 1

0x(f)h(f)ϕ(f)

n df =

∫ 1

0x(f)h(f)ej2πfn df. (78)

By the definition of DTFT and IDTFT, we complete the proof.From the above corollary, we see immediately that y(f) at the receiver end only depends on the input

x(f) at frequency f . This implies that there is no inter-symbol interference in the frequency domain, ifwe view f as a continuum of indices. The reason is that if the information is originally in the frequencydomain and modulated onto the eigenfunction {ϕ(f)

n using IDTFT, these eigenfunctions will not change theirfrequency after passing through a LTI filter, and hence in the frequency domain there will be no ISI.

However, with IDTFT as pre-processing and DTFT as post-processing (illustrated in Figure 9), weare forced to roll back to analog communication, because both the input u(f) and the output V (f)are continuous-frequency “waveforms”. In the following, let us introduce a discretized version of theDTFT/IDTFT pair called Discrete Fourier Transform (DFT), which gives us a digital implementationof OFDM.

4.2 Discrete Fourier Transform (DFT)

Discrete Fourier Transform (DFT) and the corresponding Inverse Discrete Fourier Transform (IDFT) canbe viewed as the discretized version of DTFT and IDTFT. Let us first define DFT and IDFT.



{h`}

{Zm}

{um}

{Vm}

IDTFT

DTFT

u(f)

V (f)

um =

Z 1/2

�1/2u(f)ej2⇡mf dfIDTFT:

DTFT: V (f) =X

m

Vme�j2⇡mf

Figure 9: Pre-processing and post-processing based on the DTFT/IDTFT pair

Definition 5: For a length-N sequence {xn}N−1n=0 , its (N -point) DFT is another length-N sequence {x[k]}N−1

k=0

defined as follows:

x[k] ≜ 1√N

N−1∑n=0

xne−j2π kn

N , k = 0, 1, ..., N − 1. (79)

Proposition 6: {x[k]}N−1k=0 is the N -point DFT of {xn}N−1

n=0 if and only if

xn =1√N

N−1∑k=0

xkej2π kn

N , n = 0, 1, ..., N − 1. (80)

{xn}N−1n=0 is called the Inverse Discrete Fourier Transform (IDFT) of {x[k]}N−1

k=0 .Exercise 3. Prove Proposition 6.

In this context, all the sequences are restricted to length N . Hence, the linear convolution operation isnot well defined since linear convolution of two finite-length sequences will create a longer sequence. It turnsout the right convolution operation for the DFT/IDFT world is the circular convolution defined below.Definition 7: For two length-N sequence {xn}N−1

n=0 and {hℓ}N−1ℓ=0 , their (N -point) circular convolution is

defined as follows:

(h⊛ x)n ≜N−1∑ℓ=0

hℓ x(n−ℓ) mod N , n = 0, 1, ..., N − 1. (81)

See Figure 10 for illustration of circular convolution.

The eigenfunction property (Theorem 3) and the convolution-multiplication property (Corollary 4) ofthe DTFT/IDTFT pair can be similarly obtained for the DFT/IDFT pair, summarized as follows.Theorem 8: Consider a length-N sequence {ϕ[k]

n }N−1n=0 , where ϕ

[k]n ≜ exp(j2π k

N n). It is an eigenfunction toany length-N circular convolution, that is, for any {hℓ}N−1

ℓ=0 ,(ϕ[k] ⊛ h

)n=√Nh[k]ϕ[k]

n , (82)

where h[k] is the N -point DFT of {hℓ}.



h0

h1

h2

hN�1

xN�1

x0

x1

x2

n = 0h0

h1

h2

hN�1

xN�1

x0

x1

x2

n = 2

Figure 10: Circular convolution

Proof By definition,

(ϕ[k] ⊛ h

)n=

N−1∑ℓ=0

hℓ exp(

j2π k

N[(n− ℓ) mod N ]

)(83)

(a)=

N−1∑ℓ=0

hℓ exp(

j2π k

N(n− ℓ)

)(84)

=

(N−1∑ℓ=0

hℓ e−j2π k

Nℓ

)exp

(j2π k

Nn

)=√Nh[k]ϕ[k]

n . (85)

Here (a) is due to the fact that exp(j2π k

N (n− ℓ))= exp

(j2π k

N (n− ℓ+ iN))

for any integer i ∈ Z.Corollary 9: Consider three length-N sequences {yn}, {hn}, and {xn} with yn = (x⊛ h)n. Then, N -pointDFT of them satisfy y[k] =

√Nh[k]x[k].

Exercise 4. Use Theorem 8 to prove Corollary 9.

4.3 Realizing circular convolution in LTI channel by adding cyclic prefix

We have introduced DFT/IDFT, the discretized version of DTFT/IDTFT, and shown that the right convo-lution operation there is circular convolution, not the ordinary linear convolution experienced in frequencyselective channels. Hence, suppose one would like to replace the IDTFT and DTFT blocks in Figure 9 byIDFT and DFT, Corollary 9 cannot apply.

A quick fix to the above issue is to “implement” circular convolution in an LTI channel by introducinga slight overhead called cyclic prefix (CP). Note that the difference between linear convolution and circularconvolution is the “wrap-around” operation introduced by the modulo-N operation in the index. Hence, ifwe can “tile” the two length-N sequences periodically into ranges of indices outside 0 ∼ (N − 1), do thelinear convolution, and extract the output sequence within the indices ranging from 0 to N − 1, the resultwill be the output of the circular convolution operation.

For our purpose, since the range the indices of hℓ is from 0 to L − 1 and typically L ≪ N , it sufficeto do the tiling just “once” for the input sequence {um}. In particular, we can duplicate the last L − 1symbols uN−L+1, ..., uN−1 and pad it in front of u0. This is called “cyclic prefix”. See Figure 11 and 12 forillustration.



h0 h1 hL�1· · ·

u0uN�1 · · · · · ·uN�L+1

uN�1 · · · uN�L+1

cyclic prefix

Figure 11: Cyclic prefix

transmit

uN�L+1

uN�1

u0

u1

uN�1

u0

u1

uN�1

CP

x1

xL�1

xL

xN+L�1

xL+1

add cyclic prefix

convolution

receive

y1

yL�1

yL

yL+1

yN+L�1 vN�1

v1

v0

remove cyclic prefix

vN�1

v1

v0

vm = (h~ u)m, m = 0, ..., N � 1

h0

hL�1

Figure 12: Implementation of circular convolution over LTI channel



4.4 Linear algebraic view of DFT/IDFT and circular convolution

In the discussions above, we give a signal processing view of DFT/IDFT and circular convolution. Note thatN -point DFT/IDFT and N -point circular convolution involve only length-N sequences, there is a naturallinear algebraic view of these transforms and operation.

For DFT/IDFT, let x ≜[x0 x1 · · · xN−1

]⊺ ∈ CN×1 denote the N -dimensional column vector com-prises elements of the length-N sequence {xn}N−1

n=0 . Similarly, represent the N -point DFT of sequence {xn},{x[k]}N−1

k=0 as a N -dimensional vector x. Then, both DFT and IDFT can be viewed as matrix multiplications:

DFT : x = ΦHxIDFT : x = Φx

(86)

where Φ ∈ CN×N with the (n, k)-th element

(Φ)n,k ≜ 1√N

exp(

j2πknN

), n, k ∈ {0, ..., N − 1}. (87)

Exercise 5. Show that the matrix Φ is unitary, that is, Φ−1 = ΦH.

The above exercise is equivalent to proving Proposition 6. Equation (86) provides a linear algebraicalternative to Definition 5. Similarly, there is an alternative definition of circular convolution (Definition 7):

∀n = 0, ..., N − 1, yn = (x⊛ h)n ⇐⇒ y = hcx, (88)

where

hc ≜

h0 hN−1 · · · h2 h1

h1 h0 hN−1 h2

... h1 h0. . . ...

hN−2. . . . . . hN−1

hN−1 hN−2 · · · h1 h0

∈ CN×N . (89)

Note that each column of hc is the cyclic one-step down shift of its previous column, and the first (column-0)is h =

[h0 h1 · · · hN−1

]⊺. Such a matrix is called a circulant matrix. You can easily show that each rowof a circulant matrix is the cyclic one-step right shift of its previous row.

The following theorem is an alternative to Theorem 8 and Corollary 9.Theorem 10: For any circulant matrix hc ∈ CN×N , any k-th column of the IDFT matrix Φ, denoted byϕ[k], k ∈ {0, ..., N − 1}, is an eigenvector of hc with eigenvalue

√Nh[k], that is,

hcϕ[k] =

√Nh[k]ϕ[k], ∀ k = 0, ..., N − 1. (90)

Hence, as a corollary, any circulant matrix hc ∈ CN×N has the following eigenvalue decomposition:

hc = ΦΛhΦH, (91)

where Λh ≜ diag(h[0], h[1], ..., h[N − 1]).

The above theorem tells that any circulant matrix can be diagonalized by the Discrete Fourier Trans-form basis {ϕ[k] | k = 0, 1, ..., N − 1}. This is a great advantage in ISI mitigation: the Tx pre-processingblock and the Rx post-processing block are invariant to the underlying frequency selective channel. Theonly parameter need to be well understood from the channel is the effective number of taps L. In practice,L is taken as the largest possible number of taps in the communication scenario of interest. For example,when the communication is built over DSL wires, the L is taken to be 100 ∼ 150.



4.5 OFDM system implementation

The end-to-end system diagram of a digital OFDM system is depicted in Figure 13.

N-ptIDFT

u[0]

u[1]

u[N � 1]

u0

u1

uN�1

Insert CP

uN�L+1

uN�1

u0

u1

uN�1

P/S {xn}N+L�1n=1

{h`}

{Zm}

S/PDelete CP

{Yn}N+L�1n=1

V0

V1

VN�1

V0

V1

VN�1

N-ptDFT

V [N � 1]

V [0]

V [1]

Figure 13: OFDM system diagram

The end-to-end relationship between the input and the output can be easily derived. Let us take thelinear algebraic view.

V = hcu+Z = ΦΛhΦHu+Z =⇒ ΦHV = ΛhΦ

Hu+ΦHZ =⇒ V = Λhu+ Z. (92)

Since ΦH is an unitary matrix and Z comprises i.i.d. CN (0, N0) elements, so does Z. As a result, OFDMcreates effectively N parallel “sub-channels”:

V [k] = h(fk)u[k] + Z[k], k = 0, 1, ..., N − 1 (93)

where h(·) denotes the DTFT (not DFT) of the LTI filter channel {hℓ}L−1ℓ=0 , and fk ≜ k

N , k = 0, ..., N − 1.The effective noises {Z[k] | k = 0, 1, ..., N − 1} are i.i.d. CN (0, N0). In words, the k-th sub-channel has acomplex channel coefficient h( k

N ) =√Nh[k] and AWGN noise with variance N0.

We can now better see why the inter-symbol interference comes from frequency selectivity in an OFDMsystem. Let us take a look at the channel coefficient of the k-th sub-channel, h( k

N ). Recall that the effectivediscrete-time LTI filter impulse response

hℓ = (hb ∗ g)(ℓT ) ≡ ha(ℓT ), ℓ = 0, 1, ..., L− 1. (94)

Here for notational convenience we introduce ha(t) ≜ (hb ∗ g)(t) with CTFT ha(f). One can immediatelyrecognize that the DTFT h(f) is a periodic signal with period 1, and for −1/2 ≤ f ≤ 1/2, h(f) = ha(f/T ).In other words, OFDM uses N “subcarriers” centered at the carrier frequency fc with subcarrier spacing2WN = 1

NT .Note that the N subcarriers are not equally good: some are in better condition than others. In principle,

one should send data at a higher rate in the sub-channels with higher received SNR. The best power allocation



can be found through standard convex optimization techniques and obtain the so-called “water-filling” powerallocation policy.

Let us close this section with some discussions on the OFDM system parameter. OFDM provides anelegant solution to create end-to-end ISI-free communication channels, with the cost of inserting a length-(L− 1) cyclic prefix (CP). Hence, the CP overhead can be defined as

CP Overhead ≜ L− 1

N. (95)

To make the overhead as small as possible, the system designer would like to have N , the number ofsubcarriers, as large as possible. On the other hand, N cannot be arbitrarily large. This is due to apractical constraint between different terminals called “frequency offset” due to clock asynchrony betweenTx and Rx. As a result, the subcarrier spacing cannot be too small. Recall that

Subcarrier Spacing ≜ 2W

N, (96)

where 2W is the “total” operational bandwidth.Remark. Typically, N is taken as a power of 2, because in this case, N -point DFT/IDFT can be done withcomplexity Θ(N log2N) using Fast Fourier Transform (FFT) algorithm.Remark. One of the most challenging issue of OFDM system in practice is that, peak-to-average ratio(PAR) is much higher than single-carrier systems. Hence, it requires a large dynamic range of the linearcharacteristic of the transmit power amplifier (PA), or the Tx needs to reduce its average transmit energy.As a result, much research over the past 20 years focuses on how to reduce PAR for OFDM systems.

References

[1] R. G. Gallager, Stochastic Processes, Theory for Applications. Cambridge University Press, 2013.

[2] D. Brandwood, “A complex gradient operator and its application in adaptive array theory,” IEE Pro-ceedings F - Communications, Radar and Signal Processing, vol. 130, no. 1, pp. 11–16, February 1983.

[3] M. Tomlinson, “New automatic equaliser employing modulo arithmetic,” Electronic Letters, vol. 7, no. 5,pp. 138–139, March 1971.

[4] H. Harashima and H. Miyakawa, “Matched-transmission technique for channels with intersymbol inter-ference,” IEEE Transactions on Communications, vol. 20, no. 4, pp. 774–780, August 1972.


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