Time-Domain Analysis of Discrete-Time Systems Part 2: Convolution and Properties
The discrete-time convolution integral has various properties we which review in detail. Also, the total system response of a system is solved for and various discrete-time system properties are discussed.
Commutative and Distributive Property of DT Convolution
Running Time: 5:13
This video proves the commutative and distribute properties of discrete-time convolution.
Associative and Time-Shifting Property of DT Convolution
Running Time: 11:20
This video derives the associate and time-shifting properties of discrete-time convolution. These properties are derived using simple definitions and algebraic manipulations.
Convolution with an Impulse and Width Property of DT Convolution
Running Time: 5:31
This example examines what happens when you convolve two finite-width discrete-time signals. In this case, the result is also finite-width and its width is easy to compute. Also, discrete-time convolution with a discrete-time impulse function is examined.
Discrete-Time Convolution Sum Example
Running Time: 7:54
This example performs the convolution of two "short" discrete-time signals. Since each signal is only a few samples long, the easiest way to work this problem is by directly "plugging in" to the discrete-time convolution summation equation.
Reflect, Shift, and Sum Convolution Example #1
Running Time: 10:31
This video works an example of discrete-time convolution using the "reflect, shift, and sum" approach. Basically, this means we sketch the intermediate signals within the argument of the summation operation to understand what the signals look like for different values of the time shift k.
Reflect, Shift, and Sum Convolution Example #2
Running Time: 9:57
Discrete-time convolution of two right-sided decaying exponential signals.
Analytic Approach to Discrete-Time Convolution (i.e. Table Lookup)
Running Time: 4:13
In this video, we solve for the total response of a discrete-time difference equation. This total response is the sum of both the zero-input response and the zero-state response. The specific method used to solve for this response we call the "analytic approach" since we use tables of convolution pairs to compute the required convolution instead of computing the convolution explicitly from the definition of the convolution summation.
Total Response via Convolution
Running Time: 5:19
In this video, we solve for the total response of a discrete-time difference equation. This total response is the sum of both the zero-input response and the zero-state response. These components have been solved for in previous videos, and here we essentially "tie together" all of the pieces that have been solved for.
Discrete-Time System Properties from Impulse Response
Running Time: 15:35
This video defines and examines properties of discrete-time linear systems. These properties include memoryless, causal, stable, and invertible. Each of these properties can be determined by examining/analyzing the system impulse response.