## Discrete-Time Signals and Systems​

###### We discuss sampling continuous-time signals to yield a discrete-time signal in a way to preserve all of the original signals information.  Developing these concepts requires understanding the spectrum of an impulse-sampled signal.  We also consider the spectrum of a sampled sinusoid to gain intuition on aliasing which leads to Sampling Theorem.

Introduction

5/15/14

Running Time: 3:41

This set of lectures discusses sampling of continuous-time signals.  This introduction video outlines the different topics that will be covered (i.e. sampling, reconstruction, etc.)

Intuition

5/15/14

Running Time: 5:55

Several "cartoon" examples are sketched and sampled to gain intuition on "good" ways to sample a continuous-time signal and "bad" ways to sample a continuous-time signal.  These ambiguous terms of "good" and 'bad" are made mathematically rigorous in the subsequent videos (i.e. "good" means sampling at the Nyquist rate or greater while "bad" means sampling at less than the Nyquist rate and aliasing occurs)

Fourier Transform of an Impulse Sampled Signal

5/15/14

Running Time: 9:58

We investigate impulse sampling in the frequency domain, i.e. we derive an expression for the Fourier Transform (FT) of a signal that has been impulse sampled.  If x(t) is the continuous-time signal with corresponding FT X(w), the impulse sampled version of x(t) has a FT that consists of an infinite collection of X(w) shifted up and down the frequency axis.  Each shifted version of X(w) occurs at an integer multiple of the sampling frequency ws.

Impulse Sampling

5/15/14

Running Time: 7:28

We examine a specific example of impulse sampling.  Assuming x(t) has a Fourier Transform X(w) that looks like a "triangle", we sketch the spectrum of the impulse sampled x(t) for different values of sampling frequency ws.  We see that when ws is less than 2W, aliasing occurs.

Sampling a Sinusoid (Theory)

5/15/14

Running Time: 13:05

We work a specific problem where we impulse sample a sinusoid.  We first derive an expression for the spectrum of the impulse sampled sinusoid for an arbitrary sampling rate ws.  We then sketch the impulse sampled spectrum for different values of ws and note that when ws is less than the Nyquist rate, aliasing occurs.  A simple equation for computing the aliased frequency is also presented.

Sampling a Sinusoid (Matlab)

5/15/14

Running Time: 4:54

This example also considers impulse sampling a sinusoid (similar to the previous video), but all analytic work is performed in Matlab.  The example demonstrates the concept of aliasing and how the aliased frequency can be computed.

Zero Order Hold Sampling

5/16/14

Running Time: 7:13

Zero order hold (ZOH) sampling is another method for sampling a continuous-time signal.  A ZOH sampler can be modeled as multiplication by an infinite impulse train (i.e. an impulse sampler) followed by a low-pass filter.  This low-pass filter attenuates images at multiples of the sampling frequency, but distorts the spectrum of the original continuous-time signal.

The Sampling Theorem

5/16/14

Running Time: 8:30

The Sampling Theorem tells use the rate at which we must sample a continuous-time signal if we want to preserve all signal information (i.e. avoid aliasing).  The Nyquist rate is the lowest sampling rate that avoids aliasing and is equal to 2 times the largest signal frequency, i.e. fs = 2*fm.

Required Sampling Examples

5/19/14

Running Time: 6:06

This video works two different problems where we use the sampling theorem to determine a condition on the sampling period Ts to correctly sample the given signal (e.g. Ts must be less than or equal to "x").

Ideal Reconstruction

5/19/14

Running Time: 5:31

Reconstruction is the process of obtaining the original continuous-time signal x(t) from its impulse-samples xdelta(t).  In the frequency domain, this ideal reconstruction process can be visualized as a low-pass filter that perfectly rejects all images in the impulse-sampled spectrum.  The next video works through this ideal reconstruction process in the time-domain.

Ideal Band-Limited Interpolation (i.e. Sinc Interpolation)

5/20/14

Running Time: 8:56

This video explains a time-domain version of the reconstruction process.  Reconstruction of an impulse sampled signal requires an ideal low-pass filter.  The impulse response of an ideal low-pass filter is a sinc function.  Thus, the reconstructed signal is a superposition of time-shifted and weighted since functions which is often called sinc interpolation.

Practical Reconstruction

5/20/14

Running Time: 4:14

The previous video explained how to reconstruct x(t) from an impulse sampled signal.  This video explains how to recover x(t) from its zero order hold (ZOH) samples.  In general, this requires a filter that rejects all spectrum images and perfectly inverts any distortion introduced by the original sampling filter.

Sampling and Reconstructing Signals In Matlab

5/20/14

Running Time: 6:09

This 13-video sequence on sampling ends with a Matlab example.  A continuous-time signal is sampled and reconstructed using both impulse sampling and zero-order hold sampling.  Various plots of the original signal spectrum, sampled signal spectrum, and reconstructed time-domain signal are made to describe the overall sampling and reconstruction process.