A look at the frequency response of sample-and-hold (aka boxcar filter)

boxcar_test.m

%% Frequency response of a boxcar filter
%
% A look at the frequency response of sample-and-hold.
%
% Tobin Fricke 2012-10-18

Fs = 2^16;           % Higher sample rate
R = 32;              % upsampling factor
N = 1 * Fs;          % number of sample data to generate

tau = (1/Fs) * R;    % lower sampling period

my_boxcar = ones(R, 1)/R;

% Compute the transfer function by applying the filter to random data
x = randn(N, 1);
y = filter(my_boxcar, 1, x);
nfft = N/16;
[Txy, f] = tfestimate(x, y, hanning(nfft), nfft/2, nfft, Fs);

% Analytic form, for comparison
H = sinc(R*f/Fs) .* exp(-2i*pi*f*tau/2);   

subplot(2,1,1)
semilogx(f, db(Txy), '.',...
         f, db(H), '-');

ylim([-60 10]);
legend('numerical (tfestimate)', 'analytic (sinc+delay)')
title(sprintf('Frequency response of %dX sample-and-hold', R));

subplot(2,1,2);
semilogx(f, unwrap(angle(Txy))*180/pi, ...     
         f, unwrap(angle(H) * 180/pi));

%xlim([0 (Fs/R/2)]);     % look at only the in-band region
set(gca,'ytick', (-10:1:1) * 360);

orient landscape
print -dpdf boxcar_test.pdf