# Why are there comb-like hills in a periodogram?

I am playing with the periodogram of MATLAB. I created a simple script to observe how it behaves:

rng(1);  %# initialize the random number generator

Fs = 1000;  %# Sampling frequency
duration = 0.1; %# seconds

A = 1; %# Sinusoid amplitude
f = 150; %# Sinusoid frequency
eps = 0.01;

t = 0:1/Fs:duration;
x = A * sin(2*pi*f*t) + eps * randn(size(t));

periodogram(x,[],1024,Fs);


I have no problem with the code and can write my own periodogram function using the algorithms given in the documentation but I wonder the theoretical reason behind the comb-like hills which are not 150 Hz. What do I get those instead of getting a single spike over 150 Hz? Is there anything special in the distances of the peaks of these hills?

## migrated from stackoverflow.comNov 24 '11 at 21:07

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I'm not entirely satisfied by Itamar Katz's answer, so here's my explanation.

The DFT of an $N$ length complex signal, $x[n]=e^{\imath 2\pi f n/N}$ is

$$X[k]=\mathcal{F}\{x[n]\}=\frac{e^{\imath 2\pi (f-k)}-1}{e^{\imath 2\pi (f-k)/N}-1}$$

So, the power or the magnitude squared response is given by

$$\left\vert X[k]\right\vert^2 = \left(\frac{\sin\left(\pi(f-k)\right)}{\sin\left(\pi(f-k)/N\right)}\right)^2$$

As you can see, the above expression is zero whenever $f-k$ is an integer. You can convince yourself that the denominator is zero at only one point, and at this point, taking limits gives you a value $N^2$ for the ratio. Hence, there is no point at which the expression blows up.

Now when you take the log of the above expression, $log_{10}(0)$ is $-\infty$ (or for that matter, in any base) and hence you get nulls everywhere you had a zero. This is what results in the "comb like hills" in your plot.

Here's a short illustration in Mathematica:

Clear@X
X[f_, n_] := (Sin[π (f - #)]/Sin[π (f - #)/n])^2 &
Plot[X[3, 10][k], {k, -5, 5}, PlotRange -> All]


Frequency is on the x-axis and power (linear) is on the y-axis. You can see that the zeros occur at integer values and the peak is at 3, which is the frequency I had chosen. Now taking $log_{10}$ of the above, you get nulls which give rise to the comb like structure

Here's another example with a larger $N$, showing more nulls.

A single spike (as you call it) appears theoretically only for a infinite-length sinusoid. Since your signal is 100 samples length, it is not infinite. You actually multiplied your infinite signal with a window which has a value of 1 over 100 samples, and 0 elsewhere. Since multiplication in time domain is equivalent to convolution in the frequency domain, your spectrum is a convolution of the single spike and the frequency response of the window (btw it is called rectangular window). This is the function you got.