Review of DSP
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Signal and Systems:

Signal are represented mathematically as
functions of one or more independent
variables.

Digital signal processing deals with the
transformation of signal that are discrete in
both amplitude and time.

Discrete time signal are represented
mathematically as sequence of numbers.
2
Signals and Systems:

A discrete time system is defined
mathematically as a transformation or
operator.

y[n] = T{ x[n] }
x [n]
T{.}
y [n]
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Linear Systems:

The class of linear systems is defined by the
principle of superposition.
T{x1[n]  x2[n]}  T{x1[n]}  T{x2[n]}  y1[n]  y2[n]

And

T {ax[n]}  aT {x[n]}  ay[n]
Where a is the arbitrary constant.
 The
first property is called the additivity property
and the second is called the homogeneity or scaling
property.
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Linear Systems:

These two property can be combined into
the principle of superposition,
T{ax1[n]  bx2 [n]}  aT{x1[n]}  bT{x2 [n]}
x1[n]
H
x2 [n]
H
y1[n]
y 2 [ n]
ax1[n]  bx2 [n]
Linear System
H
ay1[n]  by2 [n]
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Time-Invariant Systems:

A Time-Invariant system is a system for
which a time shift or delay of the input
sequence cause a corresponding shift in
the output sequence.
x1[n]
H
x1[n  n0 ]
y1[n]
y1[n  n0 ]
H
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LTI Systems:


A particular important class of systems consists
of those that are linear and time invariant.
LTI systems can be completely characterized by
their impulse response.
 

y[n]  T   x[k ] [n  k ]
k  


From principle of superposition:
y[ n] 

 x[k ]T  [n  k ]
k  

Property of TI:
y[n] 

 x[k ]h[n  k ]
k  
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LTI Systems (Convolution):
y[n] 

 x[k ]h[n  k ]
k  

Above equation commonly called convolution
sum and represented by the notation
y[n]  x[n]  h[n]
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Convolution properties:

Commutativity:
x[ n]  h[ n]  h[ n]  x[ n]

Associativity:
(h1[n]  h2 [n])  h3[n]  h1[n]  (h2 [n]  h3[n])

Distributivity:
h[n]  (ax1[n]  bx2[n])  a(h[n]  x1[n])  b(h[n]  x2[n])

Time reversal:
y[n]  x[n]  h[n]
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…Convolution properties:

If two systems are cascaded,
H1
H2
 The
overall impulse response of the combined
system is the convolution of the individual IR:
h[n]  h1[n]  h2 [n]
 The
overall IR is independent of the order:
H2
H1
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Duration of IR:

Infinite-duration impulse-response (IIR).

Finite-duration impulse-response (FIR)
y[n]  b0 x[n]  b1 x[n  1]  ...  bq x[n  q]

In this case the IR can be read from the
right-hand side of:
h[n]  bn
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Transforms:
Transforms are a powerful tool for
simplifying the analysis of signals and of
linear systems.
 Interesting transforms for us:

 Linearity
applies:
T [ax  by ]  aT [ x]  bT [ y ]
 Convolution
is replaced by simpler operation:
T [ x  y ]  T [ x]T [ y ]
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…Transforms:

Most commonly transforms that used in
communications engineering are:
 Laplace
transforms (Continuous in Time & Frequency)
 Continuous
 Discrete
Z
Fourier transforms (Continuous in Time)
Fourier transforms (Discrete in Time)
transforms (Discrete in Time)
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The Z Transform:

Definition Equations:
 Direct
Z transform
X ( z) 

 x[n]z
n
n  
 The
Region Of Convergence (ROC) plays an
essential role.
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The Z Transform

(Elementary functions)
:
Elementary functions and their Z-transforms:
 Unit impulse: x[ n]   [ n]
X ( z) 

  [ n] z
n
1
ROC : z  0
n  
 Delayed
X ( z) 

unit impulse: x[n]   [n  k ]
  [n  k ]z
n
z
k
ROC : z  0
n  
15
The Z Transform
 Unit
n0
 1,
u[n]  
0, otherwise
Step:

(…Elementary functions)
1
X ( z)   z 
1
1 z
n 0
n
 Exponential:

ROC : | z | 1
x[n]  a nu[n]
1
X ( z)   a z 
1
1  az
n 0
n n
:
ROC : | z || a |
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Z Transform (Cont’d)

Important Z Transforms
Region Of Convergence
(ROC)
Whole Page
Whole Page
|z| > 1
|z| > |a|
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The Z Transform

(Elementary properties)
:
Elementary properties of the Z transforms:
 Linearity:
ax[n]  by[n]  aX ( z)  bY ( z)
w[n]  x[n]  y[n]
 Convolution:
if
,Then
W ( z )  X ( z )Y ( z )
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The Z Transform
 Shifting:
(…Elementary properties)
:
x[n  k ]  z X ( z )
k
 Differences:

Forward differences of a function,
x[n]  x[n  1]  x[n]

Backward differences of a function,
x[n]  x[n]  x[n  1]
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The Z Transform
 Since
(…Region Of Convergence for Z transform)
:
x[n]  x[n]   [n  1]   [n]
the shifting theorem
Z x[n]  ( z 1) X ( z)
Z x[n]  (1  z ) X ( z )
1
20
The Z Transform
(Region Of Convergence for Z transform)
:

The ROC is a ring or disk in the z-plane
centered at the origin :i.e.,

The Fourier transform of x[n] converges at
absolutely if and only if the ROC of the
z-transform of x[n] includes the unit circle.

The ROC can not contain any poles.
21
The Z Transform
(…Region Of Convergence for Z transform)
:

If x[n] is a finite-duration sequence, then
the ROC is the entire z-plane, except
possibly z  0 or z   .

If x[n] is a right-sided sequence, the ROC
extends outward from the outermost finite
pole in X (z ) to z   .

The ROC must be a connected region.
22
The Z Transform
(…Region Of Convergence for Z transform)
:

A two-sided sequence is an infinite-duration
sequence that is neither right sided nor left sided.

If x[n] is a two-sided sequence, the ROC will
consist of a ring in the z-plane, bounded on the
interior and exterior by a pole and not containing
any poles.

If x[n] is a left-sided sequence, the ROC extends in
ward from the innermost nonzero pole in X (z ) to
z 0 .
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The Z Transform

(Application to LTI systems)
:
We have seen that y[n]  x[n]  h[n]

By the convolution property of the Z transform
Y ( z)  X ( z)H ( z)


Where H(z) is the transfer
function of system.
Stability
is stable if a bounded input | x[n] | M
produced a bounded output, and a LTI system
is stable if:
| h[k ] |  
 A system

k
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Fourier Transform
Time
Continuousaperiodic
Discreteaperiodic
Frequency
Continuousaperiodic
Continuousperiodic
Continuousperiodic
Discreteaperiodic
Discreteperiodic
Discreteperiodic
Transform Type
Fourier Transform
Discrete Time Continuous Frequency FT
Fourier Series
Discrete Time Discrete Frequency FT
25
Discrete-time Fourier Transform
j
X (e ) 

 x[n]e
 jn
n  
The same as Z-transform with z on the unit circle
Continuous in Frequency, periodic with period = 2*pi
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The Discrete Fourier Transform (DFT)

Discrete Fourier transform
N 1
X [k ]   x[n]e
n 0
 It
is customary to use the
 Then the direct form is:
N 1
 j 2kn
N
WN  e
X [k ]   x[n]W
n 0
j 2
N
 nk
N
27
The Discrete Fourier Transform (DFT)
 With
the same notation the inverse DFT is
1 N 1
nk
x[n]   X [k ]WN
N k 0
28
The DFT (Elementary functions):

Elementary functions and their DFT:
 Unit impulse: x[ n]   [ n]
X [k ]  1
 Shifted
unit impulse: x[n]   [n  p ]
X [k ]  W
 kp
N
29
The DFT (…Elementary functions):
 Constant:
x[n]  1
X [k ]  N [k ]
 Complex
exponential:
x[n]  e jn
N 

X [ k ]  N  k 

2 

30
The DFT (…Elementary functions):
 Cosine
function:
x[n]  cos 2f 0 n
N
X [k ]   [k  Nf 0 ]   [ N  k  Nf 0 ]
2
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The DFT

(Elementary properties)
:
Elementary properties of the DFT:
 Symmetry:
,Then
f [ n]  F [ k ]
f [k ]  NF[n]
 Linearity:
,Then
If
if
and
x[n]  X [k ]
y[n]  Y [k ]
ax[n]  by[n]  aX [k ]  bY [k ]
32
The DFT
(…Elementary properties)
:
 Shifting:
because of the cyclic nature of DFT
domains, shifting becomes a rotation.
if
x[n]  X [k ]
x
[(
n

p
)
]

W
N
,Then
 Time
if
,Then
 kp
N
X [k ]
reversal:
x[n]  X [k ]
x[(n) N ]  X [(k ) N ]
33
The DFT
(…Elementary properties)
:
 Cyclic
convolution: convolution is a shift,
multiply and add operation. Since all shifts in
the DFT are circular, convolution is defined
with this circularity included.
N 1
x[n]  y[n]   x[ p] y[( n  p) N ]
p 0
34