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Summary: Discrete-time systems allow for mathematically specified processes like the difference equation.
A discrete-time signal
snsn
is delayed by
n0n0
samples when we write
sn−n0snn0, with
n0>0n00. Choosing
n0n0
to be negative advances the signal along the integers. As
opposed to analog
delays, discrete-time delays can
only be integer valued. In the frequency
domain, delaying a signal corresponds to a linear phase shift of
the signal's discrete-time Fourier transform:
sn−n0↔e−(i2πfn0)Sei2πf↔snn02fn0S2f.
Linear discrete-time systems have the
superposition property.
A discrete-time system is called shift-invariant
(analogous to time-invariant analog systems) if
delaying the input delays the corresponding output.
If
Sxn=ynSxnyn, then a shift-invariant system has the property
Sxn−n0=yn−n0Sxnn0ynn0
(2)
We use the term shift-invariant to emphasize that delays can only have
integer values in discrete-time, while in analog signals, delays can
be arbitrarily valued.
We want to concentrate on systems that are both linear and
shift-invariant. It will be these that allow us the full power of
frequency-domain analysis and implementations. Because we have no
physical constraints in "constructing" such systems, we
need only a mathematical specification. In analog systems, the
differential equation specifies the input-output relationship in the
time-domain. The corresponding discrete-time specification is the
difference equation.
Here, the output signal
ynyn
is related to its past values
yn−lynl,
l=1…pl1…p, and to the current and past values of the input signal
xnxn.
The system's characteristics are determined by the choices for the
number of coefficients pp and
qq and the coefficients' values
a1…apa1…ap
and
b0b1…bqb0b1…bq.
Aside:
There is an asymmetry in the coefficients:
where is
a0a0? This coefficient would multiply the
ynyn
term in Equation 3. We have essentially
divided the equation by it, which does not change the
input-output relationship. We have thus created the convention
that
a0a0 is always one.
As opposed to differential equations, which only provide an
implicit description of a system (we must somehow
solve the differential equation), difference equations provide an
explicit way of computing the output for any
input. We simply express the difference equation by a program that
calculates each output from the previous output values, and the
current and previous inputs.
Difference equations are usually expressed in software with
for loops. A MATLAB program that would
compute the first 1000 values of the output has the form
for n=1:1000
y(n) = sum(a.*y(n-1:-1:n-p)) + sum(b.*x(n:-1:n-q));
end
An important detail emerges when we consider making this program
work; in fact, as written it has (at least) two bugs. What input
and output values enter into the computation of
y1y1? We need values for
y0y0,
y-1y-1, ..., values we have not yet computed. To compute
them, we would need more previous values of the output, which we
have not yet computed. To compute these values, we would need
even earlier values, ad infinitum. The way out of this
predicament is to specify the system's initial
conditions: we must provide the
pp output values that
occurred before the input started. These values can be
arbitrary, but the choice does impact how the system responds to
a given input. One choice gives rise to a
linear system: Make the initial conditions zero. The reason
lies in the definition of a linear system: The only
way that the output to a sum of signals can be the sum of the
individual outputs occurs when the initial conditions in each
case are zero.
Exercise 1
The initial condition issue resolves making sense of the
difference equation for inputs that start at some index.
However, the program will not work because of a programming,
not conceptual, error. What is it? How can it be "fixed?"
Solution
The indices can be negative, and this condition is not
allowed in MATLAB. To fix it, we must start the signals
later in the array.
Let's consider the simple system having
p=1p1
and
q=0q0.
yn=ayn−1+bxnynayn1bxn
(4)
To compute the output at some index, this difference equation
says we need to know what the previous output
yn−1yn1
and what the input signal is at that moment of time. In more
detail, let's compute this system's output to a unit-sample
input:
xn=δnxnδn. Because the input is zero for negative indices, we
start by trying to compute the output at
n=0n0.
y0=ay-1+by0ay-1b
(5)
What is the value of
y−1y1? Because we have used an input that is zero for all
negative indices, it is reasonable to assume that the output
is also zero. Certainly, the difference equation would not
describe a linear system if the input that is
zero for all time did not produce a zero
output. With this assumption,
y-1=0y-10, leaving
y0=by0b. For
n>0n0, the input unit-sample is zero, which leaves us with
the difference equation
∀n,n>0:yn=ayn−1nn0ynayn1. We can envision how the filter responds to this
input by making a table.
yn=ayn−1+bδnynayn1bδn
(6)
Table 1
nn
xnxn
ynyn
−11
00
00
00
11
bb
11
00
baba
22
00
ba2ba2
:
00
:
nn
00
banban
Coefficient values determine how the output behaves. The
parameter bb can be any value,
and serves as a gain. The effect of the parameter
aa is more complicated (Table 1). If it equals zero, the
output simply equals the input times the gain
bb. For all non-zero values of
aa, the output lasts forever;
such systems are said to be IIR
(Infinite Impulse
Response). The reason for this
terminology is that the unit sample also known as the impulse
(especially in analog situations), and the system's response
to the "impulse" lasts forever. If
aa is positive and less than one,
the output is a decaying exponential. When
a=1a1, the output is a unit step. If
aa is negative and greater than
−11,
the output oscillates while decaying exponentially. When
a=−1a1, the output changes sign forever, alternating
between bb and
−bb.
More dramatic effects when
|a|>1a1;
whether positive or negative, the output signal becomes larger
and larger, growing exponentially.
Figure 1:
The input to the simple example system, a unit sample, is
shown at the top, with the outputs for several system
parameter values shown below.
Positive values of aa are used in
population models to describe how population size increases
over time. Here, nn might
correspond to generation. The difference equation says that
the number in the next generation is some multiple of the
previous one. If this multiple is less than one, the
population becomes extinct; if greater than one, the
population flourishes. The same difference equation also
describes the effect of compound interest on deposits. Here,
nn indexes the times at which
compounding occurs (daily, monthly, etc.),
aa equals the compound interest
rate plus one, and
b=1b1
(the bank provides no gain). In signal processing
applications, we typically require that the output remain bounded for
any input. For our example, that means that we restrict
|a|<1a1
and choose values for it and the gain according to the application.
Exercise 2
Note that the difference
equation,
yn=a1yn−1+…+apyn−p+b0xn+b1xn−1+…+bqxn−qyna1yn1…apynpb0xnb1xn1…bqxnq
does not involve terms like
yn+1yn1
or
xn+1xn1 on the equation's right side. Can such terms also
be included? Why or why not?
Solution
Such terms would require the system to know what future
input or output values would be before the current value was
computed. Thus, such terms can cause difficulties.
Figure 2: The plot shows the unit-sample response of a length-5
boxcar filter.
Example 2
A somewhat different system has no "aa"
coefficients. Consider the difference equation
yn=1q(xn+…+xn−q+1)yn1qxn…xnq1
(7)
Because this system's output depends only on current and previous
input values, we need not be concerned with initial conditions. When
the input is a unit-sample, the output equals
1q1q
for
n=0…q−1n0…q1, then equals zero thereafter. Such systems are said
to be FIR (Finite
Impulse Response)
because their unit sample responses have finite
duration. Plotting this response (Figure 2) shows that the unit-sample response is a pulse
of width qq and height
1q1q.
This waveform is also known as a boxcar, hence the name
boxcar filter given to this system. We'll derive
its frequency response and develop its filtering
interpretation in the next section. For now, note that the
difference equation says that each output value equals the
average of the input's current and
previous values. Thus, the output equals the running average
of input's previous qq
values. Such a system could be used to produce the average
weekly temperature (q=7q7) that could be updated daily.
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"Electrical Engineering Digital Processing Systems in Braille."