Sampling - Revision history

Smitry at 21:50, 8 October 2006

2006-10-08T21:50:51Z

← Older revision		Revision as of 14:50, 8 October 2006
Line 1:		Line 1:
			*[[Signals and systems\|Signals and Systems]]
	==Introduction==		==Introduction==
	In this chapter we will look at sampling and how it affects signal processing. Sampling is the process of taking a continuous stream of data with infinite points of resolution and getting that data into a form that can be stored in a finite data set.		In this chapter we will look at sampling and how it affects signal processing. Sampling is the process of taking a continuous stream of data with infinite points of resolution and getting that data into a form that can be stored in a finite data set.

Greg: /* Sampling Overview */

2005-11-08T19:24:00Z

Sampling Overview

← Older revision		Revision as of 12:24, 8 November 2005
Line 7:		Line 7:
	<math> \bold x(n) = x_a(n*T_s) </math>		<math> \bold x(n) = x_a(n*T_s) </math>

	[[Image:Signals_Sampling.JPG]]~~greg~~		[[Image:Signals_Sampling.JPG]]

	There are, however, a few things that we need to keep in mind when trying to sample signals with many frequencies in them. When we sample data, we want to be able to sample enough data to get a correct reconstruction (this is discussed further below), but we don’t want to sample too much data that our sets become unmanageable.		There are, however, a few things that we need to keep in mind when trying to sample signals with many frequencies in them. When we sample data, we want to be able to sample enough data to get a correct reconstruction (this is discussed further below), but we don’t want to sample too much data that our sets become unmanageable.

Greg: /* Sampling Overview */

2005-11-08T19:23:50Z

Sampling Overview

← Older revision		Revision as of 12:23, 8 November 2005
Line 7:		Line 7:
	<math> \bold x(n) = x_a(n*T_s) </math>		<math> \bold x(n) = x_a(n*T_s) </math>

	[[Image:Signals_Sampling.JPG]]		[[Image:Signals_Sampling.JPG]]greg

	There are, however, a few things that we need to keep in mind when trying to sample signals with many frequencies in them. When we sample data, we want to be able to sample enough data to get a correct reconstruction (this is discussed further below), but we don’t want to sample too much data that our sets become unmanageable.		There are, however, a few things that we need to keep in mind when trying to sample signals with many frequencies in them. When we sample data, we want to be able to sample enough data to get a correct reconstruction (this is discussed further below), but we don’t want to sample too much data that our sets become unmanageable.

Wilspa: /* Sampling Theory */

2005-10-03T04:58:54Z

Sampling Theory

← Older revision		Revision as of 21:58, 2 October 2005
Line 18:		Line 18:

	Going back to our example of audio, we know that humans can hear from about 20 Hz up to about 20,000 Hz. Therefore we know that we want to make sure that we sample fast enough that we will get this entire frequency spectrum. In order to do this, we apply the principle where F is our highest frequency and T is our sampling rate to get a value of T = 40,000 Hz. This is also known as the Nyquist Theorem. This explains why audio that is recorded onto a CD is sampled at a rate of 44,100 Hz (they sample slightly higher than necessary for a 20 KHz max frequency).		Going back to our example of audio, we know that humans can hear from about 20 Hz up to about 20,000 Hz. Therefore we know that we want to make sure that we sample fast enough that we will get this entire frequency spectrum. In order to do this, we apply the principle where F is our highest frequency and T is our sampling rate to get a value of T = 40,000 Hz. This is also known as the Nyquist Theorem. This explains why audio that is recorded onto a CD is sampled at a rate of 44,100 Hz (they sample slightly higher than necessary for a 20 KHz max frequency).
	Something to note here is the fact that even though we want to sample our signal for frequencies between 20 and 20,000 Hz, most signals have many more frequencies in them due to white noise. In order to make sure that only the desired frequency is ~~samples~~ you can run your signal through a pre-filter (sometimes called an anti-aliasing filter). This guarantees that the sampled data system receives analog signals having a frequency spectrum no greater than those frequencies allowed by the filter.		Something to note here is the fact that even though we want to sample our signal for frequencies between 20 and 20,000 Hz, most signals have many more frequencies in them due to white noise. In order to make sure that only the desired frequency is sampled you can run your signal through a pre-filter (sometimes called an anti-aliasing filter). This guarantees that the sampled data system receives analog signals having a frequency spectrum no greater than those frequencies allowed by the filter.

	To see how the sampled data is reconstructed, refer to my section on how a [[User:santsh\|CD Player]] works.		To see how the sampled data is reconstructed, refer to my section on how a [[User:santsh\|CD Player]] works.

Wilspa: /* Sampling Overview */

2005-10-03T04:54:17Z

Sampling Overview

← Older revision		Revision as of 21:54, 2 October 2005
Line 9:		Line 9:
	[[Image:Signals_Sampling.JPG]]		[[Image:Signals_Sampling.JPG]]

	There are, however, a few things that we need to keep in mind when trying to sample signals with many frequencies in them. When we sample data, we want to be able to sample enough data to get a correct reconstruction, but we don’t want to sample too much data that our sets become unmanageable.		There are, however, a few things that we need to keep in mind when trying to sample signals with many frequencies in them. When we sample data, we want to be able to sample enough data to get a correct reconstruction (this is discussed further below), but we don’t want to sample too much data that our sets become unmanageable.

	==Sampling Theory==		==Sampling Theory==

Wilspa: /* Sampling Overview */

2005-10-03T04:52:07Z

Sampling Overview

← Older revision		Revision as of 21:52, 2 October 2005
Line 9:		Line 9:
	[[Image:Signals_Sampling.JPG]]		[[Image:Signals_Sampling.JPG]]

	~~Unfortunately~~, ~~there are~~ a few things that we need to keep in mind when trying to sample signals with many frequencies in them. When we sample data, we want to be able to sample enough data to get a correct reconstruction, but we don’t want to sample too much data that our sets become unmanageable.		There are, however, a few things that we need to keep in mind when trying to sample signals with many frequencies in them. When we sample data, we want to be able to sample enough data to get a correct reconstruction, but we don’t want to sample too much data that our sets become unmanageable.

	==Sampling Theory==		==Sampling Theory==

Santsh: /* Sampling Process */

2004-12-10T23:52:35Z

Sampling Process

← Older revision		Revision as of 16:52, 10 December 2004
Line 35:		Line 35:


	<math> x(n) = x_a(~~n*T_s~~) </math>		<math> \bold x(n) = x_a(nT_s) </math>


	Principle author of this page: [[User:santsh\|Shawn Santana]]		Principle author of this page: [[User:santsh\|Shawn Santana]]

Santsh: /* Sampling Process */

2004-12-10T23:52:04Z

Sampling Process

← Older revision		Revision as of 16:52, 10 December 2004
Line 35:		Line 35:


	<math> x(n) = x_a(~~nT_s~~) </math>		<math> x(n) = x_a(n*T_s) </math>


	Principle author of this page: [[User:santsh\|Shawn Santana]]		Principle author of this page: [[User:santsh\|Shawn Santana]]

Santsh: /* Sampling Process */

2004-12-10T23:51:40Z

Sampling Process

← Older revision		Revision as of 16:51, 10 December 2004
Line 33:		Line 33:

	After which the sampled signal is converted into a discrete-time signal by mapping the impulses that are spaced in time by <math> T_s </math> into a sequence x(n) where the sample values are indexed by the integer variable n:		After which the sampled signal is converted into a discrete-time signal by mapping the impulses that are spaced in time by <math> T_s </math> into a sequence x(n) where the sample values are indexed by the integer variable n:


	<math> x(n) = x_a(nT_s) </math>		<math> x(n) = x_a(nT_s) </math>

Santsh: /* Sampling Process */

2004-12-10T23:51:07Z

Sampling Process

← Older revision		Revision as of 16:51, 10 December 2004
Line 26:		Line 26:
	In order to see how the sampling process works, imagine a periodic sequence of impulses,		In order to see how the sampling process works, imagine a periodic sequence of impulses,

	<math> s_a(t) = \sum_{n=-\infty^\infty} \delta(t-nT_s) </math>		<math> s_a(t) = \sum_{n=-\infty}^\infty \delta(t-nT_s) </math>

	Now multiply your signal by the impulses to form the sampled signal		Now multiply your signal by the impulses to form the sampled signal

	<math> x_s(t) = x_a(t) * s_a(t) = \sum_{n=-\infty^\infty} x_a(nT_s)*\delta(t-nT_s) </math>		<math> x_s(t) = x_a(t) * s_a(t) = \sum_{n=-\infty}^\infty x_a(nT_s)*\delta(t-nT_s) </math>

			After which the sampled signal is converted into a discrete-time signal by mapping the impulses that are spaced in time by <math> T_s </math> into a sequence x(n) where the sample values are indexed by the integer variable n:

			<math> x(n) = x_a(nT_s) </math>


	Principle author of this page: [[User:santsh\|Shawn Santana]]		Principle author of this page: [[User:santsh\|Shawn Santana]]