Discrete-Time Signals and Systems Part 2: Signal Operations
We continue discussing basics of discrete-time signals and systems. The following videos examine a variety of discrete-time signal operations (e.g. amplitude scaling, time-shifting, time-scaling, etc.) as well as start introducing ways to represent discrete-time systems via difference equations and block diagrams.
Discrete-Time Signals Introduction
Running Time: 2:07
This is the first video in a 14-part series that continues to introduce basic concepts of discrete-time signals and systems. This introductory video outlines the basic topics that will be covered in the series to include:
1) Discrete-Time Signal Operations
2) Common Discrete-Time Signal Types
3) Discrete-Time System Examples and Representation
Signal Operations: Amplitude Scaling
Running Time: 3:29
The first signal operation we investigate is amplitude scaling. Consider the discrete-time signal x[k] and constant c. The discrete-time signal y[k] = cx[k] is an "amplitude scaled" version of x[k] since every value in the original discrete-time signal x[k] has been scaled by a factor "c".
This video shows several examples of performing amplitude scaling. Specifically, given an initial discrete-time signal x[k] we also compute and plot the signals 2x[k] and -x[k].
Signal Operations: Addition
Running Time: 4:38
The next signal operation we investigate is signal addition. Consider the discrete-time signals x1[k] and x2[k] c. The discrete-time signal y[k] = x1[k] + x2[k] is the addition of the two signal and is created by adding the two signals at each point in time k.
This video shows an several examples of performing signal addition. One uses a "graphical approach" while the second example uses a "table approach" to create the final signal.
Signal Operations: Differencing and Summation
Running Time: 7:55
This video investigates discrete-time signal differencing and summation
The discrete-time signal y[k] = x[k]-x[k-1] is the difference of adjacent time samples of the x[k]. This discrete-time operation is analogous to taking the derivative of a continuous-time signal. This operation is a high-pass operation since adjacent samples that are close in amplitude have a small difference (i.e. low-frequency terms are rejected) while adjacent samples that are different in amplitude have a large difference (i.e. high-frequency terms are amplified).
The discrete-time signal y[k] = sum x[m] for m = -infinity to k is the summation of the signal x[k]. At any time k, the signal y[k] is the cumulative sum of all previous values of x[k]. This discrete-time operaiton is analogous to taking the integral of a continuous-time signal. This operation is a low-pass operation since high-frequency changes in the signal x[k] are "smoothed out" by the summation operation.
Signal Operations: Time Scaling, Downsampling, and Upsampling
Running Time: 7:46
This video investigates discrete-time signal time scaling which includes downsampling and upsampling operations.
The time scaled signal y[k] = x[ak] is created by replaced the discrete-time variable k with ak. If a is greater than 1 we downsample the discrete-time signal x[k]. If a is between 0 and 1 we upsample (or expand) the discrete-time signal x[k].
Two different specific examples are worked. We show a downsampling example for the case of a = 2, and we show an upsampling example for the case of a = 1/2. In the upsampling case we also show two different options; either zero-insertion or linear interpolation can be used for the expanded signal values.
Signal Operations: Time Reversal
Running Time: 4:16
This video investigates discrete-time signal time time reversal.
The time reversed signal y[k] = x[-k] is created by replacing the discrete-time variable k with -k. This operation results in x[k] being "flipped" about the time origin. In addition to this basic definition of time reversal, two different specific examples are also worked through to show how to compute the time reversed version of a discrete time signal.
Signal Operations: Time Shifting
Running Time: 3:59
This video investigates discrete-time signal time shifting.
The time-shifted signal y[k] = x[k-m] is created by replacing the discrete-time variable k with k-m. This operation results in x[k] being shifted along the time axis by an amount m. When m is greater than 0, this is a shift to the right. When m is less than 0, this is a shift to the left. Several different examples of time-shifting are worked in the video.