Linear, ShiftInvariant Discrete Time systems

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Linear, Shift-Invariant Discrete Time systems

• Chapter 5

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Linear, Shift-Invariant Discrete Time systems
First, lets consider discrete time signals. A
discrete time signal is a signal which only
exists at discrete times. This is often a
continuous-time signal which is sampled at
discrete times.
One familiar discrete time signal is a movie. If
the original scene is a continuous-time signal, a
movie of that scene is a sequence of samples.
Each sample is an image of the scene at the
instant it was taken (photographed). The movie
image does not exist between samples.
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Linear, Shift-Invariant Discrete Time systems
That covers the discrete time part. What about
the linear, shift invariant parts? A
continuous-time system has a system transfer
function, H(s), which is the Laplace transform of
the system impulse response, h(t). The system is
completely described by either H(s) or h(t). If
either of those functions is known, no other
information is needed (no other information which
cant be derived from H(s) or h(t)).
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Linear, Shift-Invariant Discrete Time systems
For any input function x(t), the resulting output
y(t) can be found
or
where is the convolution operator. The system
is linear if, for any two time functions x1(t)
and x2(t), and any two constants k1 and k2,
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Linear, Shift-Invariant Discrete Time systems
This has a couple of implications. First,
multiplying the input function by a constant
results in a response which is multiplied by the
same constant (doubling the input amplitude
doubles the output amplitude)
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Linear, Shift-Invariant Discrete Time systems
Second, if the input function is a sum of two
other functions, such as u(t) - u(t-2), the
response to the sum is the sum of the responses
to the individual functions. In other words,
superposition works
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Linear, Shift-Invariant Discrete Time systems
This definition of linearity works for discrete
time systems, too. The variable t is replaced by
n
where is the discrete time convolution
operator. The system is linear if, for any two
functions x1n and x2n, and any two constants
k1 and k2,
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Linear, Shift-Invariant Discrete Time systems
Again, multiplying the input function by a
constant results in a response which is
multiplied by the same constant (doubling the
input amplitude doubles the output amplitude)
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Linear, Shift-Invariant Discrete Time systems
And if the input function is a sum of two other
functions, such as un un-2, the response to
the sum is the sum of the responses to the
individual functions (superposition works)
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Linear, Shift-Invariant Discrete Time systems
That covers the linear part. Now for the shift
invariant part. A continuous-time system is
characterized by its system impulse response
h(t), which is simply the response (output) to an
impulse (input). If the input is a delayed
impulse, d(t-k), the response is a delayed
impulse response, h(t-k)
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Linear, Shift-Invariant Discrete Time systems
If y(t) is the response of the system to some
arbitrary input function x(t), then the response
of the system to that same function delayed by k
seconds is the same function y(t), but delayed by
k seconds
For a time-invariant system, delaying the input
merely delays the response by the same amount it
doesnt change the shape of the output waveform
or its amplitude.
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Linear, Shift-Invariant Discrete Time systems
The discrete-time equivalent of time invariance
is called shift invariance, because t is replaced
with n which is simply the position of a sample
in a sequence. Changing n shifts the sample one
way or the other. For a shift invariant system,
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Difference equations
Continuous time linear, shift invariant (LSI)
systems are described by linear differential
equations with constant coefficients.
Discrete time LSI systems (dont confuse this
usage of the acronym LSI with its other
meaning, large-scale integration) are described
by linear difference equations with constant
coefficients
This says the present value of the output is a
weighted sum of past values of the input and
output.
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IIR and FIR systems
A system based on this general difference
equation, which depends on past outputs
(feedback) as well as inputs (feedforward), is
called an Infinite Impulse Response (IIR) system.
Its impulse response never quite decays to
zero, so it is infinite in duration.
A Finite Impulse Response (FIR) system includes
only feedforward terms its present output
depends only on present and past inputs.
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IIR and FIR systems
Lets consider an example of a second-order IIR
system. The difference equation is
A block diagram implementing this system is shown
in the next slide.
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IIR and FIR systems
b0
X(n)
S
Z-1
-a1
y(n-1)
b1
Z-1
Z-1
X(n-1)
-a2
Z-1
y(n-2)
b2
X(n-2)
z-1 represents a delay of one sample period
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IIR and FIR systems
This can be implemented in Matlab very easily, as
shown in the next two slides
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IIR example
If the coefficients and the input sequence are
a 0, -0.5, 0 b 1, 0, 0 x 1, 1, 1,
1, 1, 1, 1, 1, 1, 1
and we run the Matlab example, it is apparent
that
By the way, x(n) u(n), so y(n) is the step
response. The results are shown in the next
slide
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IIR example
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IIR example
This example is causal, meaning that the response
cannot preceed the input which produces it. All
real-world systems must be causal! Its output
is at rest until the input is applied, at n 0.
We can demonstrate this, (and shift invariance at
the same time by running it again with x
0, 0, 0, 0, 0, 1, 1, 1, 1, 1
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IIR example
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Discrete Time Impulse
Well apply the sequence d(n) to the IIR filter
m-file (next slide) Its apparent that
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Discrete Time Impulse