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DSPA 301 Lecture Notes 5
Discrete Correlation
Correlation is a measure of similarity between two signals and is found using a process similar to
convolution. The algorithm is also used to cancel system noise.
The discrete cross-correlation (denoted **) of x[n] and h[n] is defined by:
rxh [n]  x[n] * *h[n] 
rhx [n]  h[n] * *x[n] 


k  
k  


 x[k ]h[k  n]   x[k  n].h[n]
 h[k ]x[k  n] 
k  
 h[k  n].x[k ]
k  
Correlation is the convolution of one signal with the flipped version of the other.
rxh [n]  x[n] * *h[n]  x[n] * h[n]
rhx [n]  h[n] * *x[n]  h[n] * x[ n]
Autocorrelation
The correlation rxx [n] of a signal x[n] with itself is called autocorrelation. In terms of the convolution
operation the equation is described as:
rxx [n]  x[n] * *x[n]  x[n] * x[ n]
Autocorrelation is an even symmetric function with
rxx [n]  rxx [ n]
Autocorrelation has a maximum at n= 0 and satisfies the inequality
| rxx [n] | rxx [0]
Auto correlation can never exceed its value at the origin. Correlation is effective in detection of noise in
signals and systems. Noise is uncorrelated with the signal, if correlation is applied the signal with itself,
the correlation will only be due to the signal showing a sharp peak at zero.
Stability of LTI Systems
System Stability is an important practical constraint in filter design.
Bounded input, bounded output stability (BIBO)
If the inputs remain bounded, the weighted sum of the inputs is also bounded. Conditions for BIBO
stability involves the roots of its characteristic equation. Every root of its characteristic equation must
have a magnitude less than unity, i.e. every root “r” must have magnitude | r | 1 (less than unity). The
impulse response must be absolutely summable with:
S

| h[k ] | 
k  
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DSPA 301 Lecture Notes 5
Example: A filter described by the impulse response h[ n]  (0.5) u[n] has a difference equation
n
described by y[n]  x[n]  ay[n  1] . Determine if the system is stable.


h  
n 0
 | h[n] |  (0.5) n 
The system is stable
1
2
1  0.5
2
Example: Consider the system described by
y[n] 
1
1
y[n  1]  y[n  2]  x[n] . Determine if the
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system is stable.
The system is stable since the roots of its characteristic equation have a magnitude | r | 1 .
z2 
1
1
1
1
z   0 the roots are z1  and z 2  
6
6
2
3
Examples:
1. A filter is described by y[n]  x[n  1]  x[n] which has the impulse response h[n]  {1,  1} . It is
a non-causal because h[n]   [n  1]   [n] is not zero for
n  0 . Determine if the system is stable.
2. Determine if the following system is stable: y[n]  x[n]  2 x[n  1] .
3. Determine if the following is stable: y[n]  2 y[n  1]  y[n  2]  x[n]
Z-Transform Analysis of Discrete Systems
The z-transform for a discrete system is described as:
For the sequence: x[n]  x[0], x[1], x[2], x[3],..... is described by

X ( z )   x[n]. z n
n 0
Expanding the summation: X ( z )  x[0]z 0  x[1]z 1  x[2]z 2  x[3]z 3  ......
Example: For the sequence
x[n]  {3, 2, 5, 8, 4} the z-transform of the finite sequence is
X ( z )  3 z 0  2 z 1  5 z 2  8 z 3  4 z 4
Examples: Find the z-transform of:
1.  [n]
2. u[n]
3. x[ n]  (0.5) u[ n]
n

X ( z )   x[n]. z
n 0
n


 (0.5)
n  
n
.z
n

n
1
z
 0.5 
 

 
1
z  0.5
1  0.5 z
n 0  z 
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DSPA 301 Lecture Notes 5
Region of Convergence (ROC)
For a finite sequence x[n], the z-transform X(z) is a polynomial which is finite and converges for all z,
except z = 0. Thus, the ROC for finite sequences is the entire z-plane, except perhaps for z = 0 and/or
z = ∞, as applicable. If X(z) is a rational function in z, its ROC depends on the one or two-sidedness of
x[n], as illustrated in Figure below. The ROC excludes all pole locations (denominator roots) where X(z)
becomes infinite. As a result, the ROC of right-sided signals is |z| > |p|max and lies exterior to a circle
of radius |p|max, the magnitude of the largest pole. The ROC of causal signals, with x[n] = 0, n < 0,
excludes the origin and is given by 0 > |z| > p|max. Similarly, the ROC of a left-sided signal x[n] is |z| <
|p|min and lies interior to a circle of radius |p|min, the smallest pole magnitude of X(z). The ROC of a
two-sided signal x[n] is |p|min < |z| < |p|max, an annulus whose radii correspond to the smallest and
largest pole locations in X(z). We use inequalities of the form |z| < |α| (and not |z| ≤ |α|), for example,
because X(z) may not converge at the boundary.
The ROC of the z-Transform X(z) Determines the Nature of the Signal x[n] Finite-length x[n]: ROC of
X(z) is all the z-plane, except perhaps for z = 0 and/or z = ∞. Right-sided x[n]: ROC of X(z) is outside a
circle whose radius is the largest pole magnitude. Left-sided x[n]: ROC of X(z) is inside a circle whose
radius is the smallest pole magnitude. Two-sided x[n]: ROC of X(z) is an annulus bounded by the largest
and smallest pole radius.
Example: Let
X ( z) 
z
z

Find the ROC of the system.
z2 z3
Its ROC depends on the nature of the system. For the above system:

If the signal is assumed right-sided, the ROC is |z| > 3 (because |p|max = 3).

If the signal is left-sided, the ROC is |z| < 2 (because |p|min = 2).

If assumed two-sided, the ROC is 2 < |z| < 3.
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DSPA 301 Lecture Notes 5
Examples:
1. Consider the following system
2. If Y ( z ) 
3.
G( z) 
X ( z) 
z  0.5
, what is the ROC?
z
z 1
, what is the ROC if the Signal y[n] is right –sided?
( z  0.1)( z  0.5)
z
z

, the ROC of the system is two-sided, find the roots of the system and
z  2 z 1
sketch the ROC.
4.
V ( z) 
( z  3)
, Find the and sketch the ROC if v[n] is left sided.
( z  2)( z  1)
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DSPA 301 Lecture Notes 5
Z-Transform Table