Remarks on DFT

The two parameters that we have control over in the DFT are thesampling interval ∆t and the total number of points N thatdetermine the sampling period N∆t.

The sampling interval ∆t determines the range of frequencies thatcan be resolved (up to fNyquist = 1

2∆t ), while N determines thedetails that can be resolves (∆f = 1

N∆t )

Let’s look at some examples to see how the choice of ∆t and N∆taffects the DFT.

Example 1: What happens if the sample rate is too low? i.e. thesignal contains frequencies that are larger than the Nyquistfrequency.

Aliasing

Consider a sine function f (t) = sin(2πft), with f = 4Hz ,∆t = 0.2s and N = 32. fNyquist = 1/(2 · 0.2s) = 2.5 Hz. Therefore,f > fNyquist

Aliasing

Below is the power spectrum of the sine function.

There is a peak at 1 Hz and 4 Hz, where the latter corresponds tothe negative frequency −1 Hz.

Aliasing

The frequency f = 4 Hz is 1.5 Hz above the Nyquist frequency(2.5 Hz). It gets folded back into the frequency range below theNyquist frequency. This is called aliasing.Due to the symmetry YN/2−n = Y ∗N/2+n around the Nyquistfrequency, it appears as a low frequency fNyquist − 1.5 Hz = 1 Hz.This is what we see in the power spectrum.

Note: When sampling audio data, for example, a low pass filter isused to cut off frequencies beyond the audible spectrum of 20 KHzin order to avoid aliasing. The filter rolls off smoothly for a fewkHz. Therefore, CD’s audio are typically sampled at2 · 22050 = 44100 Hz after the filter applied.

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Aliasing

Aliasing can be illustrated if we overplot a 1 Hz sinusoid onto thesampling points. On the ∆t = 0.2 sampling interval, 1 and 4 Hzfrequencies are indistinguishable.

DFT - comments I

When frequencies higher than 1/1∆t are present in a signalthat undergoes DFT, aliasing occurs. Typically this leads tofolding back of high frequency components to lowerfrequencies.

When sampling a signal at a certain sampling rate 1/∆t itmay be desirable to pass it through a low pass filter to get ridof frequencies above fNyquist.

When going from frequency back to the time domain, oftenget non-zero imaginary components, while the real part isreconstructed correctly.

Spectral/Frequency Leakage

Consider a triangular wave signal with period T = 3.2 s sampled at∆t = 0.1 s. The total number of sampling points is N = 500.Therefore, the total sampling period is 50 s.

Spectral/Frequency Leakage

Below is the computed Power spectrum |DFT (f (t))|2. Expect a aseries of equally spaced sharp peaks at integer multiples of thef = 1/T = 0.3125 Hz. However, peaks are broad!

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Spectral/Frequency Leakage

Let’s increase the sampling period to 51.2 s, which is an integermultiple of the period of the triangular wave. The totalnumber of sampling points is now N = 512.

Spectral/Frequency Leakage

Peaks in the power spectrum (below) are sharp and occupy onlyfrequency bins at integer multiples of f .

Spectral/Frequency Leakage

Reason: DFT yields frequency spectrum at discrete frequenciesn

N∆t , with n = 0 . . .N/2− 1.

If frequency components of signal do not match thesediscrete frequencies exactly, frequency amplitudes getsmeared out. This is refereed to as “spectral leakage” or“frequency leakage”.

If the signal/function is periodic with frequency f = 1/T , we knowthat it contains only frequencies that are integer multiples of f . Bychoosing a sample period N∆t = mT , where m is an integer, weensure that frequency bins can match f exactly, so leakage isavoided.

Windowing

There is another way of mitigating the effects of spectral leakagein the power spectrum: Applying a window function to the signal.The Fourier transform of a finite sampling period can be consideredas a product between the infinite signal and the window function:

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Windowing

Multiplication of two functions in the time domain corresponds toconvolution of their Fourier transforms in the frequency domain(Convolution theorem works in both directions). The discreteFourier transform of the rectangular window is just∑N−1

m=0 e−i2πmn/N . However, in order to see the leakage of the

absolute amplitudes, one looks at the square of the Fouriertransform:

Wn =1

N2

∣∣∣∣∣N−1∑m=0

e−i2πmn/N

∣∣∣∣∣2

=1

N2

[sin(πn)

sin(πn/N)

]2

This function has a main peak and side lobes. It is the side lobesthat smear out the frequency spectrum and cause leakage. Thegoal is to find a different window function where the side lobes falloff faster as a function of the frequency bins.

Windowing

There are plenty of window functions,but these are the mostcommon ones:

wj = 1−

∣∣∣∣∣ j − 12N

12N

∣∣∣∣∣ ≡ “Bartlett window”

wj =1

2

[1− cos

(2πj

N

)]≡ “Hann window”

wj =

(1−

j − 12N

12N

)2

≡ “Welch window”

Windowing

Numerical recipes (page 560)

Windowing

Numerical recipes (page 561)

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Windowing

Example: Apply Hann window function to the triangular wave(N = 500), where period of the function is not commensurate withthe sampling period.

Windowing

Black points denote the power spectrum without the windowfunction, red points correspond to the power spectrum of theproduct of triangular wave with the window function. Peaks aremuch sharper!

DFT - comments II

If signal is periodic, the easiest way to avoid spectral leakageis to choose the sampling period to be an integer multiple ofthe period of the function.

For non-periodic signals or if periodicity is unknownwindowing can mitigate the effects of leakage. Choice ofwindow function determines trade-off between narrowness ofmain peak and the decay of the sidelobes. Rectangularwindow has narrow main peak (good to determine comparablestrength signals) but slow decay of sidelobes (bad for resolvingnearby frequency components).

Windowing can be useful when trying to resolve nearbyfrequencies that are masked/washed out by spectral leakage.

Optimal Wiener filtering

Previously we considered simple deconvolution of a signal that isnoise free. Recall the convolution theorem:

F[u ⊗ r ] = F[u]F[r ]

Therefore, if s = u ⊗ r , then

u = F−1

[F[s]

F[r ]

]However, in practice the measured signal s(t) is usually corruptedwith noise, which makes this method unreliable.

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Optimal Wiener filtering

In general, a measured signal can be written as

c(t) = [u(t)⊗ r(t)] + n(t) (1)

c(t) = s(t) + n(t), (2)

where u(t) is the noise free, deconvoluted signal, r(t) the impulseresponse function and n(t) the noise term.

Goal: Construct a filter φ(t) (or Φ(f )) to be applied to c(t) (orC (f )) in order to obtain an estimate of the uncorrupted signal u(t)(or U(f )).In particular we will estimate the true signal U(f ) as follows:

U(f ) =C (f )Φ(f )

R(f )(3)

Optimal Wiener filtering

We demand U to be close to U in the mean square sense.Therefore we want to minimize the following integral:∫ ∞

−∞|u(t)− u(t)|2dt =

∫ ∞−∞|U(f )− U(f )|2df ,

where we used Parseval’s theorem.Substituting eqn.’s (2) and (3), we obtain

=

∫ ∞−∞

∣∣∣∣C (f )Φ(f )

R(f )− S(f )

R(f )

∣∣∣∣2 df ,=

∫ ∞−∞

∣∣∣∣ [S(f ) + N(f )]Φ(f )

R(f )− S(f )

R(f )

∣∣∣∣2 df ,

Optimal Wiener filtering

=

∫ ∞−∞

∣∣∣∣C (f )Φ(f )

R(f )− S(f )

R(f )

∣∣∣∣2 df ,=

∫ ∞−∞|R(f )|−2 |[S(f ) + N(f )]Φ(f )− S(f )|2 df ,

=

∫ ∞−∞|R(f )|−2 |S(f )[Φ(f )− 1] + N(f )Φ(f )|2 df

=

∫ ∞−∞|R(f )|−2

[|S(f )|2|1− Φ(f )|2 + |N(f )|2|Φ(f )|2

]df

In the last step cross product between N and S integrated over thefrequencies gives zero, since they are uncorrelated.

Optimal Wiener filtering

In order to find Φ(f ) that minimizes the integral we differentiatethe integrand and set it to zero:

∂

∂Φ(f )

[|S(f )|2|1− Φ(f )|2 + |N(f )|2|Φ(f )|2

]= 0

−2|S(f )|2|1− Φ(f )|+ 2|N(f )|2|Φ(f )| = 0

Therefore,

Wiener Filter

Φ(f ) =|S(f )|2

|S(f )|2 + |N(f )|2(4)

which we can use to estimate true signal:

U(f ) =C (f )Φ(f )

R(f )

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Optimal Wiener filtering

Still a little problem

Φ(f ) =|S(f )|2

|S(f )|2 + |N(f )|2

The Wiener filter involves S , the smeared (convoluted signal) andthe noise N. However, we only know the measured noisy signalC (f ) = S(f ) + N(f ) (or c(t) = s(t) + n(t)).

How do we determine Φ(f ) ?

We can estimate |S(f )|2 and |S(f )|2 + |N(f )|2 from the powerspectrum of the measured signal c(t), since|C (f )|2 = |S(f )|2 + |N(f )|2.(This equation be proven via the Wiener-Khinchin theorem)

Optimal Wiener filtering

Extrapolate noise spectrum to separate |S(f )|2 and |N(f )|2(|N(f )|2 can be flat, tilted, smoothly varying - does not matter).

Optimal Wiener filtering

Remarks:

Quality of results second order in precision to which theoptimal filter is determined. Therefore, even a crudelydetermined filter (say 10 percent) can give excellent results.

For this reason separation of S and N by eye is sufficient forpractical purposes.

We do not need to know the response function r(t) (or R(f ))to find Φ(f ).

Don’t forget to extend filter Φ(f ) to negative frequenciesΦ(−f ) = Φ(f ).

Notice that Φ(f ) is close to unity where noise is negligible andclose to zero where noise is dominant. Wiener filter is theoptimal way to go between these extremes.

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