Mark's Homework 15 - Revision history

Pridma: /* Explanations */

2007-12-04T01:07:18Z

Explanations

← Older revision		Revision as of 18:07, 3 December 2007
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	==Explanations==		==Explanations==

			Actually, this homework was relatively straight-forward. I sampled a signal, and ran it through the interpolation filter. The filter works by inserting (oversampling rate) - 1 times the number of points on the original DFT of the sampled signal. This causes the signal to be interpolated as well as oversampled.

			For canceling out the effects of the D/A converter, I simply DFT'd the pulse that a D/A converter would output, inverted it, and multiplied it with the DFT of my interpolating filter. This cancels out the effects of the D/A converter. There is hardly a difference due to oversampling. When you oversample the signal, the D/A converter pulse becomes much wider, so the edges where it goes down to zero don't go down to zero anymore. If that made any sense...

Pridma: /* MATLAB Script */

2007-12-04T01:02:54Z

MATLAB Script

← Older revision		Revision as of 18:02, 3 December 2007
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	f = -1/(2T):1/(NT):1/(2T)-1/(NT);		f = -1/(2T):1/(NT):1/(2T)-1/(NT);
	plot(f,abs(fftshift(X)))		plot(f,abs(fftshift(X)))
	title('Frequency response of the ~~origianl~~ signal');		title('Frequency response of the original signal');
	xlabel('Frequency (Hz)');		xlabel('Frequency (Hz)');
	ylabel('Amplitude');		ylabel('Amplitude');

Pridma: /* Graphs */

2007-12-04T01:02:41Z

Graphs

← Older revision		Revision as of 18:02, 3 December 2007
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	==Graphs==		==Graphs==

	[[Image:OVR20071203Fig1.png\|thumb\|left\|~~694px~~\| Figure 1. The ~~sampled signal with only 7 sample points~~.]]		[[Image:OVR20071203Fig1.png\|thumb\|left\|1600px\| Figure 1. The signals.]]

	[[Image:OVR20071203Fig2.png\|thumb\|left\|694px\| Figure 2. The ~~DFT of Figure 1~~.]]		[[Image:OVR20071203Fig2.png\|thumb\|left\|694px\| Figure 2. The interpolated signal.]]

	[[Image:OVR20071203Fig3.png\|thumb\|left\|894px\| Figure 3. The ~~same as Figure 1 but with 600 sample points, and zoomed in~~.]]		[[Image:OVR20071203Fig3.png\|thumb\|left\|894px\| Figure 3. The frequency response of the original signal.]]

Pridma at 00:59, 4 December 2007

2007-12-04T00:59:24Z

Show changes

Pridma: /* MATLAB Script */

2007-12-03T23:14:40Z

MATLAB Script

← Older revision		Revision as of 16:14, 3 December 2007
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	clear all;		clear all;
	close all;		close all;
	N = 50; %Number of sample points for our original signal		N = 10; %Number of sample points for our original signal
	tmax=2; %Time, in seconds, to extend our original signal. We then sample this signal N times		tmax=2; %Time, in seconds, to extend our original signal. We then sample this signal N times
	oversampling = 4; %Number of oversampling points		oversampling = 4; %Number of oversampling points
	T=tmax/N; %Calculate T, the sample period		T=tmax/N; %Calculate T, the sample period
	t=0:T:(N)*T;		t=0:T:(N-1)*T;
	x=sin(2pit)+sin(2pi2t)+sin(2pi5t); %This is our signal to over sample		x=sin(2pit)+sin(2pi2t)+sin(2pi5t); %This is our signal to over sample
	N1=(oversampling-1)*N;		N1=(oversampling-1)*N;
Line 18:		Line 18:

	x2= ifft(X1);		x2= ifft(X1);

	~~figure(7)~~
	~~t1 = 0:N/(N+N1)T:(N-1/(N1+N))T;~~
	~~plot(t1, x2, 'bo');~~
	~~hold on~~
	~~t=0:T:(N)*T;~~
	~~plot(t, x, 'ro', t, x, 'r-');~~
	~~hold off~~

	~~f = -1/(2T):1/(NT):1/(2*T);~~

	~~figure(8)~~
	~~plot(f, abs(fftshift(x)))~~

	%This is to cancel out the effects of the D/A Converter		%This is to cancel out the effects of the D/A Converter
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	p = u(t+1e-9) - u(t-T/(2oversampling)) + u(t - (oversamplingT-T/(2*oversampling))); %This is p(t) sampled		p = u(t+1e-9) - u(t-T/(2oversampling)) + u(t - (oversamplingT-T/(2*oversampling))); %This is p(t) sampled
	P = fft(p);		P = fft(p);
			P(161) = 1e-19;
	iP = 1./P; %Take 1/P(f) so we cancel the effects of the D/A converter (which is P(f))		iP = 1./P; %Take 1/P(f) so we cancel the effects of the D/A converter (which is P(f))
	numpoints = 2*T/sT;		numpoints = 2*T/sT;
Line 62:		Line 50:
	figure(1)		figure(1)
	t1 = 0:N/(N+N1)T:(N-1/(N1+N))T;		t1 = 0:N/(N+N1)T:(N-1/(N1+N))T;
	plot(t1, x2, 'bo');		plot(t1, x1, 'bo');
	hold on		hold on
	t=0:T:(N)*T;		t=0:T:(N-1)*T;
	plot(t, x, 'ro', t, x, 'r-');		plot(t, x, 'ro', t, x, 'r-');
	hold off		hold off
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	%figure(4)		%figure(4)
	%plot(f, abs(fftshift(X1)))		%plot(f, abs(fftshift(X1)))

	</pre>		</pre>

Pridma: /* MATLAB Script */

2007-12-03T22:59:24Z

MATLAB Script

← Older revision		Revision as of 15:59, 3 December 2007
Line 7:		Line 7:
	clear all;		clear all;
	close all;		close all;
	N = ~~500~~; %Number of sample points for our original signal		N = 50; %Number of sample points for our original signal
	tmax=2; %Time, in seconds, to extend our original signal. We then sample this signal N times		tmax=2; %Time, in seconds, to extend our original signal. We then sample this signal N times
	oversampling = 4; %Number of oversampling points		oversampling = 4; %Number of oversampling points

Pridma: New page: ==Problem== Make a MATLAB script to do four times oversampling and a filter so as to eliminate as much as possible the effect of the D/A converter that follows the interpolation filter. =...

2007-12-03T22:58:22Z

New page: ==Problem== Make a MATLAB script to do four times oversampling and a filter so as to eliminate as much as possible the effect of the D/A converter that follows the interpolation filter. =...

New page

==Problem==
Make a MATLAB script to do four times oversampling and a filter so as to eliminate as much as possible the effect of the D/A converter that follows the interpolation filter.

==MATLAB Script==
The following is currently '''NON-WORKING'''. A working script will follow.
<pre>
clear all;
close all;
N = 500; %Number of sample points for our original signal
tmax=2; %Time, in seconds, to extend our original signal. We then sample this signal N times
oversampling = 4; %Number of oversampling points
T=tmax/N; %Calculate T, the sample period
t=0:T:(N)*T;
x=sin(2*pi*t)+sin(2*pi*2*t)+sin(2*pi*5*t); %This is our signal to over sample
N1=(oversampling-1)*N;
X=fft(x);
X1 = (N1+N)/N*[X(1:N/2), zeros(1, N1), X(N/2+1:N)]; %This is our interpolating filter

x2= ifft(X1);

figure(7)
t1 = 0:N/(N+N1)*T:(N-1/(N1+N))*T;
plot(t1, x2, 'bo');
hold on
t=0:T:(N)*T;
plot(t, x, 'ro', t, x, 'r-');
hold off

f = -1/(2*T):1/(N*T):1/(2*T);

figure(8)
plot(f, abs(fftshift(x)))

%This is to cancel out the effects of the D/A Converter
sT = T/100; %sT is hwo many times we are going to sample our pulse p(t)
stmax = oversampling*T; %This is how wide our pulse has to be when we oversample
sN = stmax/sT; %This is the total number of sampling points for sampling p(t)
t = 0:sT:stmax-sT;
p = u(t+1e-9) - u(t-T/(2*oversampling)) + u(t - (oversampling*T-T/(2*oversampling))); %This is p(t) sampled
P = fft(p);
iP = 1./P; %Take 1/P(f) so we cancel the effects of the D/A converter (which is P(f))
numpoints = 2*T/sT;
%We can only go out to 1/(2*T) on the 1/P(f). The rest we need to zero out.
iP = [iP(1:numpoints) zeros(1, sN-numpoints*2-1) iP(sN-numpoints:sN)]
iP = iP(1:sN/(N1+N):sN); %Down sample 1/P(f) so we can element multiply with our interpolating filter

%This plots P(f)
f = -1/(2*sT):1/(sN*sT):1/(2*sT)-1/(sN*sT);
Ps = fftshift(P); %Shift P(f)
%Figure 4 plots P(f) shifted
figure(4)
plot(f, abs(Ps));
%Figure 5 plots P(f) shifted and zoomed in the f axis
figure(5)
plot(f(sN/2-numpoints+1:sN/2+numpoints), abs(Ps(sN/2-numpoints+1:sN/2+numpoints)));

X1 = X1 .* iP; %Element multiply our interpolating filter with our D/A converter effect canceller filter
x1 = ifft(X1);

%Figure 1 plots the original signal and our interpolated signal (with D/A
%converter effects cancelled.)
figure(1)
t1 = 0:N/(N+N1)*T:(N-1/(N1+N))*T;
plot(t1, x2, 'bo');
hold on
t=0:T:(N)*T;
plot(t, x, 'ro', t, x, 'r-');
hold off

%Figure 2 plots just our interpolated signal, but this time with lines
%connecting each point.
figure(2)
plot(t1, x1, 'b-');

%Figure 3 plots the frequency response of our original sampled signal.
figure(3)
Xs((N+1)/2:N) = X(1:(N+1)/2); %The fftshift didn't work for me
Xs(1:(N+1)/2) = X((N+1)/2:N); %So this manually shifts the FFT
f = -1/(2*T):1/(N*T):1/(2*T)-1/(N*T);
plot(f,abs(Xs))
%figure(4)
%plot(f, abs(fftshift(X1)))
</pre>

==Graphs==

==Explanations==