% EE-3220 Digital Signal Processing

% Tuesday 1 December 2009

% Dr. Durant <durant@msoe.edu



% Demonstrate that, when sampled at fs = 4 Hz, 1.5 and 2.5 Hz

% cosine waves are aliased to the same values.  Nyquist tells us 

% that we will have aliasing for frequencies above fs/2 = 2 Hz.  

% Recall that the 2 signals have 2-sided spectrum frequencies 

% of -1.5 and 1.5, and -2.5 and 2.5 Hz.  They are aliased 

% about fs = 4 Hz, so we have images of the 2.5 Hz wave 

% at 4-2.5=1.5 and 4+2.5=6.5 Hz.  There are also images 

% at all integer multiples of fs, including the negative ones

% of fs, including -fs = -4 Hz, giving -4-2.5 = -6.5 Hz and

% -4+2.5 = -1.5 Hz.  We see +/-1.5 Hz are among the aliased

% images of the 2.5 Hz signal, explaining why they have the

% same samples at fs = 4 Hz.



t = linspace(0,1); % time from 0 to 1 s, default 100 samples for plenty of detail

y1_5 = cos(2*pi*1.5*t); % 1.5 Hz cosine at all times t

y2_5 = cos(2*pi*2.5*t); % 2.5 Hz cosine

plot(t,y1_5, t,y2_5) % Plot both cosines vs. time



frs = [1.5 2.5]; % put the cosine frequencies in an array for reference below

fs = 4; % sampling frequency, Hz

N = 5; % number of samples

t_samp = linspace(0, (N-1)/fs, N) % sampling time: start, finish, number of samples



hold on % add to existing plot instead of replacing it

for tv = t_samp % foreach sampling time,

	% plot circles representing sampled value for all frequencies...

	plot(tv, cos(2*pi*frs*tv), 'ko'); % black circles

end