BITS Pilani
Pilani Campus
5th Lecture DSP
IMPLEMENTATION OF DISCRETE-TIME SYSTEMS
Structures for the Realization of Linear Time-Invariant Systems
As a beginning, let us consider the first-order system
which is realized as in Fig.
direct form I structure.
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system can be viewed as two linear time-invariant systems
in cascade.
The first is a nonrecursive, system described by the
equation
second is a recursive system described by the equation
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nonrecursive
Recursive
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if we interchange the order of the recursive and nonrecursive
systems
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From this figure we obtain the two difference equations
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two elements can be merged into one delay, as shown as
direct form II structure
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direct form II structure can be generalized
for the general linear time invariant recursive system
described by the difference equation
Cascade of non recursive and recursive system
&
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N > M. This structure
is the cascade of a
recursive system
followed by a non recursive system
the direct form I1 structure requires M+N+1
multiplications and max{M, N} delays
Also called a cononic form.
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If ak = 0 for all k=1,2,.,N, Then
which is a non recursive linear time-invariant system
And is called the FIR system with impulse response h(k)
equal to the coefficients bk
For M = 0, LTI system becomes purely recursive
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second-order systems
Direct Form II realization fig.a
If we set a1=a2=0
If we set b1=b2=0
purely recursive system
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CORRELATION OF DISCRETE-TIME SIGNALS
received signal sequence as
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Digital communications
BITS Pilani, Deemed to be University under Section 3 of UGC Act, 1956
Crosscorrelation and Autocorretation Sequences
The crosscorrelation o f x(n) and y(n)is a sequence rxy(l),
which is defined as
or, equivalently. as
If we reverse the roles of x(n) and y(n)
or
By comparing we conclude that
Therefore, rxy(l) is simply the folded version of ryx(l),
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Determine the cross correlation sequence rxy (I) of
the sequences x(n) & y(n)
For l = 0 we have
The product sequence v0 (n) = x(n)y(n) is
and hence the sum over all values of n is
For I > 0. we shift y(n)to the right relative to x(n) by I
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For I < 0. we shift y(n)to the left relative to x(n) by I
Therefore. the cross correlation sequence of y(n) & x(n) is
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Properties of the Autocorrelation and
Cross correlation Sequences
Let two sequences x(n) and y(n) with finite energy
The energy in this signal
First, we note that rxx(0) = Ex and ryy(0) = Ey
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Dividing by b2
In the speciai case where y(n) = x(n)
This means that the autocorrelation sequence of a
signal attains its maximum value at zero lag
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The Schur-Cohn Stability Test
The poles of the system are the roots of the denominator
polynomial of H (z), namely
We denote a polynomial of degree m by
The reciprocal or reverse polynomial Bm(z) of degree
m is defined as
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First, we set
Then we compute the lower-degree
polynomials A, ( z ) , m = N , N - 1, N - 2,
. . . , 1,
according to the recursive equation
where the coefficients Km are defined as
has all its roots inside the unii circle if and only if the
coefficients Km sarisfy the condition Km < 1 for all m = 1, 2. .
...N
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Determine if the system having the system
function is stable
We begin with A2(z), which is defined as
Hence ; K2= -1/2
Now And
Therefore K1= -7/2
Since K1>1 the system is unstable
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