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Difference Equations 47
Note that the signal-processing toolbox in MATLAB also provides a
function called xcorr for sequence correlation computations. In its simplest
form
>> xcorr(x,y)
computes the crosscorrelation between vectors x and y, while
>> xcorr(x)
computes the autocorrelation of vector x. Itgenerates results that are
identical to the one obtained from the proper use of the conv m function.
However, the xcorr function cannot provide the timing (or lag) information
(as done by the conv m function), which then must be obtained by
some other means.
2.4 DIFFERENCE EQUATIONS
An LTI discrete system can also be described by a linear constant coefficient
difference equation of the form
N∑
M∑
a k y(n − k) = b m x(n − m), ∀n (2.21)
k=0
m=0
If a N ̸=0,then the difference equation is of order N. This equation describes
a recursive approach for computing the current output, given the
input values and previously computed output values. In practice this equation
is computed forward in time, from n = −∞ to n = ∞. Therefore
another form of this equation is
M∑
N∑
y(n) = b m x(n − m) − a k y(n − k) (2.22)
m=0
k=1
A solution to this equation can be obtained in the form
y(n) =y H (n)+y P (n)
The homogeneous part of the solution, y H (n), is given by
y H (n) =
N∑
c k zk
n
k=1
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