Using Coherence and Incoherence

Suppose we are given as data the M complex numbersdm=e^imγ, for m= 1, ..., M, and we are asked to find the real numberγ. We can exploit coherent summation to get our answer.

First of all, from the data we have been given, we cannot distinguishγ fromγ+ 2π, since, for all integersm

e^im(γ+2π)=e^imγe^2mπi=e^imγ(1) =e^imγ.

Therefore, we assume, from the beginning, that theγ we want to find lies in the interval [−π, π). Note that we could have selected any interval of length 2π, not necessarily [−π, π); if we have no prior knowledge of where γis located, the intervals [−π, π) or [0,2π) are the most obvious choices.

4.11.1 The Discrete Fourier Transform

Now we take any value ω in the interval [−π, π), multiply each of the numbersdmbye^−imω, and sum overmto get

DF T_d(ω) = M m=1

dme^−imω. (4.11)

The sum we denote byDF T_dwill be called the discrete Fourier transform (DFT) of the data (column) vectord= (d₁, ..., dM)^T. We define the column vectoreωto be

eω= (e^iω, e^2iω, ..., e^{iM ω})^T,

which allows us to writeDF T_d=e^†_ωd, where the dagger denotes conjugate transformation of a matrix or vector.

Rewriting the exponential terms in the sum in Equation (4.11), we obtain

DF T_d(ω) = M m=1

d_me^−imω = M m=1

e^im(γ−ω).

Performing this calculation for eachωin the interval [−π, π), we obtain the functionDF T_d(ω). For eachω, the complex numberDF T_d(ω) is the sum ofM complex numbers, each having length one, and angleθm=m(γ−ω).

So long asωis not equal toγ, theseθmare all different, andDF T_d(ω) is an incoherent sum; consequently,|DF T_d(ω)|will be smaller thanM. However, whenω =γ, eachθ_m equals zero, and DF T_d(ω) =|DF T_d(ω)| =M; the reason for putting the minus sign in the exponent e^−imω is so that we get the term γ−ω, which is zero when γ = ω. We find the true γ by computing the value |DF T_d(ω)| for finitely many values of ω, plot the result and look for the highest value. Of course, it may well happen that the true value ω = γ is not exactly one of the points we choose to plot;

it may happen that the true γ is half way between two of the plot’s grid points, for example. Nevertheless, if we know in advance that there is only one trueγ, this approach will give us a good idea of its value.

In many applications, the numberM will be quite large, as will be the number of grid points we wish to use for the plot. This means that the number DF T_d(ω) is a sum of a large number of terms, and that we must calculate this sum for many values ofω. Fortunately, we can use the FFT for this.

Ex. 4.4 The Dirichlet kernelof sizeM is defined as DM(x) =M

m=−Me^imx. Use Equation (4.5) to obtain the closed-form expression

D_M(x) = sin((M+¹₂)x) sin(^x₂) ; note that DM(x)is real-valued.

Ex. 4.5 Obtain the closed-form expressions M

m=N

cosmx= cos

M+N

2 x

sin(^M−₂^N+1x)

sin^x₂ (4.12)

and

M m=N

sinmx= sin

M+N

2 x

sin(^M−₂^N+1x)

sin^x₂ . (4.13)

Hint: Recall that cosmx and sinmx are the real and imaginary parts of e^imx.

Ex. 4.6 Obtain the formulas in the previous exercise using the

Ex. 4.7 Graph the functionDM(x)for various values of M.

We note in passing that the function D_M(x) equals 2M + 1 for x= 0 and equals zero for the first time atx=_2M^2π₊₁. This means that themain lobeofD_M(x), the inverted parabola-like portion of the graph centered at x= 0, crosses the x-axis atx= 2π/(2M+ 1) and x=−2π/(2M + 1), so its height is 2M + 1 and its width is 4π/(2M + 1). AsM grows larger the main lobe ofDM(x) gets higher and thinner.

In the exercise that follows we examine the resolving ability of the DFT.

Suppose we haveM equi-spaced samples of a functionf(x) having the form f(x) =e^ixγ1+e^ixγ2,

where γ₁ and γ₂ are in the interval (−π, π). If M is sufficiently large, the DFT should show two peaks, at roughly the valuesω=γ₁andω=γ₂. As the distance|γ₂−γ₁|grows smaller, it will require a larger value ofM for the DFT to show two peaks.

Ex. 4.8 For this exercise, we takeγ₁=−αandγ₂=α, for someαin the interval(0, π). Select a value ofM that is greater than two and calculate the valuesf(m)for m= 1, ..., M. Plot the graph of the function|DF T_d(ω)|on (−π, π). Repeat the exercise for various values ofM and values ofαcloser to zero. Notice howDF T_d(0)behaves asαgoes to zero. For each fixed value ofM there will be a critical value of αsuch that, for any smaller values of α,DF T_d(0)will be larger than DF T_d(α). This is loss of resolution.

4.12 Complications

In the real world, of course, things are not so simple. In most appli-cations, the data comes from measurements, and so contains errors, also callednoise. The noise terms that appear in eachdmare usually viewed as random variables, and they may or may not be independent. If the noise terms are not independent, we say that we havecorrelated noise. If we know something about the statistics of the noises, we may wish to process the data using statistical estimation methods, such as thebest linear unbiased estimator(BLUE).

4.12.1 Multiple Signal Components

It sometimes happens that there are two or more distinct values of ω that we seek. For example, suppose the data is

dm=e^imα+e^imβ,

form = 1, ..., M, whereαand β are two distinct numbers in the interval [0,2π), and we need to find bothαandβ. Now the functionDF T_d(ω) will be

DF T_d(ω) = M m=1

(e^imα+e^imβ)e^−imω = M m=1

e^imαe^−imω+ M m=1

e^imβe^−imω, so that

DF T_d(ω) = M m=1

e^im(α−ω)+ M m=1

e^im(β−ω).

So the functionDF T_d(ω) is the sum of theDF T_d(ω) that we would have obtained separately if we had had onlyαand onlyβ.

4.12.2 Resolution

If the numbersαandβ are well separated in the interval [0,2π) orM is very large, the plot of|DF Td(ω)| will show two high values, one near ω =αand one nearω =β. However, if the M is smaller or theαand β are too close together, the plot of|DF Td(ω)|may show only one broader high bump, centered betweenαandβ; this is loss of resolution. How close is too close will depend on the value ofM.

4.12.3 Unequal Amplitudes and Complex Amplitudes It is also often the case that the two signal components, the one from αand the one fromβ, are not equally strong. We could have

dm=Ae^imα+Be^imβ,

whereA > B >0. In fact, bothAandB could be complex numbers, that is,A=|A|e^iθ1 andB=|B|e^iθ2, so that

dm=|A|e^imα+θ1+|B|e^imβ+θ2.

In stochastic signal processing, theAandBare viewed as random variables;

AandB may or may not be mutually independent.

4.12.4 Phase Errors

It sometimes happens that the hardware that provides the measured data is imperfect and instead of giving us the values dm = e^imα, we get dm = e^imα+φm. Now each phase error φm depends on m, which makes matters worse than when we hadθ₁ andθ₂ previously, neither depending on the indexm.