Zero-Padding vs Increasing the Observation Size By: Prof. Dr. Erhan A. İnce Electrical and Electronic Engineering Dept. FALL 2016-2017 e-mail: erhan.ince@emu.edu.tr http://faraday.ee.emu.edu.tr/eeng420

A: Zero padding the input sequence and how it affects DFT B: Effect of Increasing window length for input x[n] on frequency resolution

Frequency Interval/Resolution Comparison DFT’s frequency resolution ∆ Fres ∼ 1/NT [Hz] and covered frequency interval ∆F = N∆Fres = 1/T = Fs [Hz]. Frequency resolution is determined only by the length of the observation interval, whereas the frequency interval is determined by the length of sampling interval.

Example (DFT Resolution): Two complex exponentials with two close frequencies F1 = 10 Hz and F2 = 12 Hz sampled with the sampling interval T = 0.02 seconds. Consider various data lengths N = 10, 15, 30, 100 with zero padding to 512 points. DFT with N = 10 and zero padding to 512 points. Not resolved: F2 − F1 = 2 Hz < 1/(NT) = 5 Hz.

DFT with N = 15 and zero padding to 512 points DFT with N = 15 and zero padding to 512 points. Not resolved: F2 − F1 = 2 Hz < 1/(NT) ≈ 3.3 Hz.

DFT with N = 30 and zero padding to 512 points DFT with N = 30 and zero padding to 512 points. Resolved: F2 − F1 = 2 Hz > 1/(NT) ≈ 1.7 Hz.

DFT with N = 100 and zero padding to 512 points DFT with N = 100 and zero padding to 512 points. Resolved: F2 − F1 = 2 Hz > 1/(NT) = 0.5 Hz.

If zero-adding does not make the DFT a better approximation of the DFTF why do we use it? There are two reasons We want more densely spaced samples of the DFT (usefull when showing it graphically) We want out input sequence length to be a power of two (for using FFT)

If zero padding does not help more accurately recover the two impulses for the input signal x[n] how can we get a better representation of the two impulses? Ans: We must widen our window of x[n]

fs = 100; T = 1/fs; N = 300; n = 0:N-1; t = n*T; x = cos (2*pi*3*t); X_mags = abs(fft(x)); subplot(2,1,1) plot(x) xlabel('Samples'); ylabel('Amplitude'); title('Time domain signal') num_disp_bins = 25; subplot(2,1,2) plot([0:num_disp_bins-1],X_mags(1:num_disp_bins),'k.'); hold on plot([0:num_disp_bins-1],X_mags(1:num_disp_bins)) hold off xlabel('Frequency Bins'); ylabel('Magnitude'); title('Frequency Magnitude')

x1= [zeros(1,362) x zeros(1,362)]; N = length(x1); n= 0:N-1; X_mags1 = abs(fft(x1)); figure subplot(2,1,1) plot(n,x1) subplot(2,1,2) plot(X_mags1 )

x22= [zeros(1,724) x zeros(1,724)]; N = length(x22); n= 0:N-1; X_mags1 = abs(fft(x22)); figure subplot(2,1,1) plot(n,x22) subplot(2,1,2) plot(X_mags1 )

fs = 100; T = 1/fs; N = 800; n = 0:N-1; t = n*T; xnew = cos (2*pi*3*t); xnew= [zeros(1,112) xnew zeros(1,112)]; N = length(xnew); n= 0:N-1; X_mags = abs(fft(xnew)); figure subplot(2,1,1) plot(n,xnew) subplot(2,1,2) plot(X_mags )

Window size 800 Padded to 1024

fs = 100; T = 1/fs; N = 1500; n = 0:N-1; t = n*T; xnew = cos (2*pi*3*t); xnew= [zeros(1,274) xnew zeros(1,274)]; N = length(xnew); n= 0:N-1; X_mags = abs(fft(xnew)); figure subplot(2,1,1) plot(n,xnew) subplot(2,1,2) plot(X_mags )

Window size 1500 Zero padded to 2048

Conclusion Zero padding does not help increase the resolution. Widening the window size (wider observation interval) helps obtain better frequency resolution.