# Discrete-Time Convolution - Department of Electrical _amp; Computer

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```					 EE313 Linear Systems and Signals          Fall 2010

Discrete-Time Convolution
Prof. Brian L. Evans
Dept. of Electrical and Computer Engineering
The University of Texas at Austin

Initial conversion of content to PowerPoint
Discrete-time Convolution
• Output y[n] for input x[n]

• Any signal can be decomposed
into sum of discrete impulses

• Apply linearity properties of

• Apply shift-invariance

• Apply change of variables
7-2
Discrete-time Convolution
• Filtering viewpoint               x[n]                y[n]

Hold impulse response h[n] in place and change variables
Flip and slide input signal x[n] about impulse response
• Example of finite impulse response (FIR) filter
Impulse response has finite extent (non-zero duration)
h[n]   Averaging filter
impulse response    y[n] = h[0] x[n] + h[1] x[n-1]
= ( x[n] + x[n-1] ) / 2
n
0      1   2     3                                           7-3
Convolution in Both Domains
• Continuous-time convolution of x(t) and h(t)

For each value of t, we compute a different (possibly)
infinite integral.
Discrete-time definition is the continuous-time definition
with integral replaced by summation

• Linear time-invariant (LTI) system
Output signal in time domain is convolution of impulse
response and input signal
Impulse response uniquely characterizes the LTI system
7-4
Convolution Demos
• Johns Hopkins University Demonstrations
http://www.jhu.edu/~signals
Convolution applet to animate convolution of simple
signals and hand-sketched signals
Convolve two rectangular pulses of same width gives a
triangle (see handout E)
• Some conclusions from the animations
Convolution of two causal signals gives a causal result
Non-zero duration (called extent) of convolution is sum of
extents of two signals being convolved minus one
7-5
Fundamental Theorem
• The Fundamental Theorem of Linear Systems
If one inputs a complex sinusoid into an LTI system, then
the output will be a complex sinusoid of the same
frequency that has been scaled by the frequency
response of the LTI system at that frequency
Scaling may attenuate the signal and shift it in phase
Example in continuous time: see handout G
Example in discrete time. Let x[n] = e j W n,

H(W) is the discrete-time Fourier transform of h[n] and
is also called the frequency response                 7-6
Frequency Response
• For continuous-time LTI system

• For discrete-time LTI system

• Note: Identity for cosine input assumes a real-
valued impulse response
7-7
Example Frequency Response
• System response to complex exponential e j W n
for all possible frequencies W in rad/sample
|H(W)|                   Ð H(W)
passband
stopband               stopband

W                      W
-Ws -Wp     Wp Ws

• Passes low frequencies, a.k.a. lowpass filter
7-8
Differentiator/Difference Operation
• Continuous                  • Discrete
f(t)            y(t)          f[n]       y[n]

We can remove scaling
by 1/Ts without changing
lowpass response
7-9
First-Order FIR Filters
x[n]

y[n] = ½ x[n] + ½ x[n-1]
y[n] = ½ x[n] - ½ x[n-1]

n                                n
Signal with a spike                  Output of
averaging filter
signal = [ 1 1 1 1 1 10 1 1 1 1 1 ];                                                               n
figure(1); stem(signal);                                                       Output of
averagingFilter = [ 0.5 0.5 ];
average = conv(averagingFilter, signal);
difference filter
figure(2); stem(average);
differenceFilter = [ 0.5 -0.5 ];
difference = conv(differenceFilter, signal);
7 - 10
figure(3); stem(difference);
Mandrill Demo (DSP First)
• First-order difference FIR filter
h[n]   First-order difference
impulse response
Highpass filter (sharpens
input signal)                                            n
Impulse response is {1, -1}
• Five-tap discrete-time (scaled) averaging FIR
filter with input x[n] and output y[n]

Lowpass filter (smooth/blur input signal)
Impulse response is {1, 1, 1, 1, 1}
7 - 11
Mandrill Demo (DSP First)
• DSP First, Ch. 6, Freq. Response of FIR Filters,
http://www.ece.gatech.edu/research/DSP/DSPFirstCD/
• From lowpass filter to highpass filter
original image ® blurred image ® sharpened/blurred image
• From highpass to lowpass filter
original image ® sharpened image ® blurred/sharpened image
• Frequencies that are zeroed out can never be
recovered (e.g. DC is zeroed out by highpass filter)
• Order of two LTI systems in cascade can be
switched under the assumption that computations
are performed in exact precision              7 - 12
Mandrill Demo (DSP First)
• Input image is 256 x 256 matrix
Each pixel represented by eight-bit number in [0, 255]
0 is black and 255 is white for monitor display
• Each filter applied along row then column
Averaging filter adds five numbers to create output pixel
Difference filter subtracts two numbers to create output pixel
• Full output precision is 16 bits per pixel
Demonstration uses double-precision floating-point data and
arithmetic (53 bits of mantissa + sign; 11 bits for exponent)
No output precision was harmed in making of this demo J
7 - 13
Linear Time-Invariant System
• Any linear time-invariant system (LTI)
system, continuous-time or discrete-time,
can be uniquely characterized by its
Impulse response: response of system to an impulse
Frequency response: response of system to a two-
sided complex exponential input signal for all
possible frequencies
Transfer function: Laplace transform (or z-transform)
of impulse response
• Given one of the three, we can find other
two provided that they exist                             7 - 14

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