Discrete Time Convolution

Signals and Systems, I Discrete Time Convolution
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Signals and Systems, I Discrete Time Convolution
Signals and systems, I
Homework: Discrete Time Convolution
INTRODUCTION
CONVOLUTION
Convolution is a very important technique in signals and systems. While continuous-time
convolution is important for theoretical analysis, you have to understand how to do discrete-
time convolution in order to write a program to implement it on a computer or a DSP chip.
This assignment will help you understand how to perform discrete time convolution operation
with delta, window, unit step and exponential functions.
For a discrete time, Linear and Time Invariant (LTI) system with impulse response h[n] as
shown in Fig. 1, the output y[n] can be obtained through the convolution of input signal x[n]
and impulse response h[n] as defined in Eqn. 1.
TASK ONE:
Convolution with a Delta Function
Calculate the convolution of the following two signals 𝑦[𝑛]=𝑥[𝑛]∗ [ℎ 𝑛]
a. Write x[n] as a sum of delta functions similar to Eqn. 6.
From the graph given:
x [ n ]=¿
Therefore:
x [ n ] =σ [ n−1 ]−σ [ n−2 ]
b. Compute convolution using the method shown in Eqn. 8. Plot your y[n] in a figure
similar to Fig. 2.
From the graph we are given that:
x [ n ] =σ [ n−1 ]−σ [ n−2 ]
h [ n ] =σ [ n ] +2 σ [ n−1 ] +−σ [ n−2 ]
From the convolution theory:
y [ n ] =x [ n ]∗h [ n ] i
h [ n ]∗σ [ n−n0 ]=σ [ n−n0 ] ii
Applying equation i and ii we obtain:

Signals and Systems, I Discrete Time Convolution
y [ n ] =h [ n ]∗σ [ n−1 ] −h [ n ]∗σ [ n−2 ]
y [ n ] =σ [ n−1 ]+ σ [ n−2 ]−3 σ [ n−3 ] +σ [ n−4 ]
This plot is as shown below:
Figure 1: plot of the resultant signal from calculation.
c. Go to http://jdsp.engineering.asu.edu/JDSP-HTML5/JDSP.html. Build the simulation
diagram as shown in Fig. 1.
Figure 2: Convolution as a sum of shifted and scaled input signals
Open up Plot 6, 7 and 9. Choose “Plot Quantity” as “Real” at the top and “Plot” as
“Discrete” at the bottom. Take a screen shot. Verify Plot 9 is a sum of Plot 6 and 7, as
y[n] is a sum of scaled and shifted h[n]’s.

Signals and Systems, I Discrete Time Convolution
Figure 3: Graph for plot 6
Figure 4: Graph for plot 7
Figure 5: Graph for plot 9