HW11 Aliasing Example - Revision history

Rothsa at 22:47, 3 December 2007

2007-12-03T22:47:34Z

← Older revision		Revision as of 15:47, 3 December 2007
Line 3:		Line 3:
	How often is often enough? Well, the Nyquist-Shannon sampling thereom says twice as much as the highest frequency. To be exact, the theorem states: "Exact reconstruction of a continuous-time baseband signal from its samples is possible if the signal is bandlimited and the sampling frequency is greater than twice the signal bandwidth."		How often is often enough? Well, the Nyquist-Shannon sampling thereom says twice as much as the highest frequency. To be exact, the theorem states: "Exact reconstruction of a continuous-time baseband signal from its samples is possible if the signal is bandlimited and the sampling frequency is greater than twice the signal bandwidth."

	So what happens when you sample as the Nyquist-Shannon thereom suggests~~, and what happens if you dont~~? I hate Matlab, but in the self-sacrificing spirit I have towards you, my dear reader, I will use everything in my power including Matlab to clarify things. ~~Hence:~~		So what happens when you don't sample as the Nyquist-Shannon thereom suggests? I hate Matlab, but in the self-sacrificing spirit I have towards you, my dear reader, I will use everything in my power including Matlab to clarify things.

			In the following picture, I have sampled the waveform at a rate higher than it's frequency, but less than twice the frequency. As you can see, the reconstructed signal is
			nowhere near the original signal.

	[[Image:sally1.jpg]]		[[Image:sally1.jpg]]

Rothsa at 22:47, 30 November 2007

2007-11-30T22:47:21Z

← Older revision		Revision as of 15:47, 30 November 2007
Line 13:		Line 13:
	sample = 1.4;		sample = 1.4;

	T = 1/sample %period;		T = 1/sample %period

	t = 0:0.01:3.14; %graph accuracy to .01;		t = 0:0.01:3.14; %graph accuracy to .01

	signal = cos(2pit); %original signal;		signal = cos(2pit); %original signal

	ts = 0:T:3.14;		ts = 0:T:3.14;
Line 25:		Line 25:
	reconstructed = cos(2pi(frequency-sample)*t);		reconstructed = cos(2pi(frequency-sample)*t);

	plot(t, signal, 'g', ts, sampledpoints, 'p',t, reconstructed, 'b');		plot(t, signal, 'g', ts, sampledpoints, 'p',t, reconstructed, 'b')

	legend('Original Signal', 'Sampled Points', 'Reconstructed Signal');		legend('Original Signal', 'Sampled Points', 'Reconstructed Signal')

	xlabel('Time');		xlabel('Time')

	ylabel('Amplitude');		ylabel('Amplitude')

	title('Aliasing Example');		title('Aliasing Example')

	So it has the same shape, but as you can see the frequency is alot lower than the original. Basically what not sampling enough does is that it recognizes multiple frequencies as the same frequency, so you lose crucial details in the reproduction of a sound.		So it has the same shape, but as you can see the frequency is alot lower than the original. Basically what not sampling enough does is that it recognizes multiple frequencies as the same frequency, so you lose crucial details in the reproduction of a sound.

Rothsa at 06:10, 30 November 2007

2007-11-30T06:10:57Z

← Older revision		Revision as of 23:10, 29 November 2007
Line 4:		Line 4:

	So what happens when you sample as the Nyquist-Shannon thereom suggests, and what happens if you dont? I hate Matlab, but in the self-sacrificing spirit I have towards you, my dear reader, I will use everything in my power including Matlab to clarify things. Hence:		So what happens when you sample as the Nyquist-Shannon thereom suggests, and what happens if you dont? I hate Matlab, but in the self-sacrificing spirit I have towards you, my dear reader, I will use everything in my power including Matlab to clarify things. Hence:

			[[Image:sally1.jpg]]

	Octave Script:		Octave Script:

Rothsa at 05:58, 30 November 2007

2007-11-30T05:58:56Z

← Older revision		Revision as of 22:58, 29 November 2007
Line 6:		Line 6:

	Octave Script:		Octave Script:

	frequency = 1;		frequency = 1;

	sample = 1.4;		sample = 1.4;

	T = 1/sample %period;		T = 1/sample %period;

	t = 0:0.01:3.14; %graph accuracy to .01;		t = 0:0.01:3.14; %graph accuracy to .01;

	signal = cos(2pit); %original signal;		signal = cos(2pit); %original signal;

	ts = 0:T:3.14;		ts = 0:T:3.14;

	sampledpoints = cos(2pifrequency*ts);		sampledpoints = cos(2pifrequency*ts);

	reconstructed = cos(2pi(frequency-sample)*t);		reconstructed = cos(2pi(frequency-sample)*t);

	plot(t, signal, 'g', ts, sampledpoints, 'p',t, reconstructed, 'b');		plot(t, signal, 'g', ts, sampledpoints, 'p',t, reconstructed, 'b');

	legend('Original Signal', 'Sampled Points', 'Reconstructed Signal');		legend('Original Signal', 'Sampled Points', 'Reconstructed Signal');

	xlabel('Time');		xlabel('Time');

	ylabel('Amplitude');		ylabel('Amplitude');

	title('Aliasing Example');		title('Aliasing Example');

	So it has the same shape, but as you can see the frequency is alot lower than the original. Basically what not sampling enough does is that it recognizes multiple frequencies as the same frequency, so you lose crucial details in the reproduction of a sound.		So it has the same shape, but as you can see the frequency is alot lower than the original. Basically what not sampling enough does is that it recognizes multiple frequencies as the same frequency, so you lose crucial details in the reproduction of a sound.

Rothsa at 05:57, 30 November 2007

2007-11-30T05:57:21Z

← Older revision		Revision as of 22:57, 29 November 2007
Line 5:		Line 5:
	So what happens when you sample as the Nyquist-Shannon thereom suggests, and what happens if you dont? I hate Matlab, but in the self-sacrificing spirit I have towards you, my dear reader, I will use everything in my power including Matlab to clarify things. Hence:		So what happens when you sample as the Nyquist-Shannon thereom suggests, and what happens if you dont? I hate Matlab, but in the self-sacrificing spirit I have towards you, my dear reader, I will use everything in my power including Matlab to clarify things. Hence:

	~~Here is a sin wave,<math>sin~~(~~2pit~~)~~</math>~~		Octave Script:
			frequency = 1;
	~~Here is that sine wave sampled at points twice it's~~ frequency		sample = 1.4;
			T = 1/sample %period;
	~~Here is that sine wave sampled at the same frequency as the wave~~		t = 0:0.01:3.14; %graph accuracy to .01;
			signal = cos(2pit); %original signal;
	~~Here is the correctly sampled sine wave reconstructed~~		ts = 0:T:3.14;
			sampledpoints = cos(2pifrequency*ts);
	~~Here is the insufficiently sampled sine wave reconstructed~~		reconstructed = cos(2pi(frequency-sample)*t);
			plot(t, signal, 'g', ts, sampledpoints, 'p',t, reconstructed, 'b');
			legend('Original Signal', 'Sampled Points', 'Reconstructed Signal');
			xlabel('Time');
			ylabel('Amplitude');
			title('Aliasing Example');

	So it has the same shape, but as you can see the frequency is alot lower than the original. Basically what not sampling enough does is that it recognizes multiple frequencies as the same frequency, so you lose crucial details in the reproduction of a sound.		So it has the same shape, but as you can see the frequency is alot lower than the original. Basically what not sampling enough does is that it recognizes multiple frequencies as the same frequency, so you lose crucial details in the reproduction of a sound.

Rothsa at 22:36, 16 November 2007

2007-11-16T22:36:07Z

← Older revision		Revision as of 15:36, 16 November 2007
Line 5:		Line 5:
	So what happens when you sample as the Nyquist-Shannon thereom suggests, and what happens if you dont? I hate Matlab, but in the self-sacrificing spirit I have towards you, my dear reader, I will use everything in my power including Matlab to clarify things. Hence:		So what happens when you sample as the Nyquist-Shannon thereom suggests, and what happens if you dont? I hate Matlab, but in the self-sacrificing spirit I have towards you, my dear reader, I will use everything in my power including Matlab to clarify things. Hence:

	Here is a sin wave,<math>sin(~~2pit~~)</math>		Here is a sin wave,<math>sin(2pit)</math>

	Here is that sine wave sampled at points twice it's frequency		Here is that sine wave sampled at points twice it's frequency

Rothsa at 22:35, 16 November 2007

2007-11-16T22:35:55Z

← Older revision		Revision as of 15:35, 16 November 2007
Line 5:		Line 5:
	So what happens when you sample as the Nyquist-Shannon thereom suggests, and what happens if you dont? I hate Matlab, but in the self-sacrificing spirit I have towards you, my dear reader, I will use everything in my power including Matlab to clarify things. Hence:		So what happens when you sample as the Nyquist-Shannon thereom suggests, and what happens if you dont? I hate Matlab, but in the self-sacrificing spirit I have towards you, my dear reader, I will use everything in my power including Matlab to clarify things. Hence:

	Here is a sin wave,		Here is a sin wave,<math>sin(2pit)</math>

			Here is that sine wave sampled at points twice it's frequency

			Here is that sine wave sampled at the same frequency as the wave

			Here is the correctly sampled sine wave reconstructed

			Here is the insufficiently sampled sine wave reconstructed

			So it has the same shape, but as you can see the frequency is alot lower than the original. Basically what not sampling enough does is that it recognizes multiple frequencies as the same frequency, so you lose crucial details in the reproduction of a sound.

Rothsa: New page: Aliasing - This is when you're sampling a function, but you don't sample often enough to reconstruct the waveform as it was before. How often is often enough? Well, the Nyquist-Shannon s...

2007-11-16T05:17:21Z

New page: Aliasing - This is when you're sampling a function, but you don't sample often enough to reconstruct the waveform as it was before. How often is often enough? Well, the Nyquist-Shannon s...

New page

Aliasing - This is when you're sampling a function, but you don't sample often enough to reconstruct the waveform as it was before.

How often is often enough? Well, the Nyquist-Shannon sampling thereom says twice as much as the highest frequency. To be exact, the theorem states: "Exact reconstruction of a continuous-time baseband signal from its samples is possible if the signal is bandlimited and the sampling frequency is greater than twice the signal bandwidth."

So what happens when you sample as the Nyquist-Shannon thereom suggests, and what happens if you dont? I hate Matlab, but in the self-sacrificing spirit I have towards you, my dear reader, I will use everything in my power including Matlab to clarify things. Hence:

Here is a sin wave,