Lecture-3:

Review of required digital signal processing

 Convolution
Convolution is one of the most frequently used operations
in DSP. Specially in digital filtering applications where two
finite and causal sequences x[n] and h[n] of lengths N1
and N2 are convolved
 
y[n] = h[n]  x[n] =  h[k ]x[n − k ] =  h[k ]x[n − k ]
k = − k =0
 Operations involved

 Folding
 Shifting
 Multiplication
 Summation

ES & BM Signal Processing 3

 
y (t ) =

 x( )h(t −  )
= −
h()


x()


y(t)
t
 
y (t ) =

 x( )h(t −  )
= −

h(-)


x()


y(t)
t
 
y (t ) =

 x( )h(t − )
= −

h(t-), t=0


x()


y(t)
t
 
y(t) =  x( )h(t −  )
 =−

h(t-), t=1



x()


y(t)
t
 
y(t) =  x( )h(t −  )
 =−

h(t-),t=2



x()


y(t)
t
 
y(t) =  x( )h(t −  )
 =−

h(t-),t=5



x()


y(t)
t
 Example convolutions
 
f *g=  f ( )g(t −  ) =  g( ) f (t −  )
 =−  =−
The sum of the product of 2 functions after 1
is reversed & shifted.

 Convolution of two rectangles

http://mathworld.wolfram.com/Convolution
 Example convolutions
 
f *g=  f ( )g(t −  ) =  g( ) f (t −  )
 =−  =−
The sum of the product of 2 functions after 1 is
reversed & shifted.

 Convolution of two Gaussians

h(n)={2,1,-3,0} x(n)={1,2,3,0}

INTRODUCTION TO DIGITAL SPEECH PROCESSING (ET60007) © CET, IIT KGP

 Circular convolution

•Circular convolution of x(n) and h(n) is defined as

the convolution of h(n) with a periodic signal xp(n) :

y p ( n) = x p ( n)  h( n)
N −1
y p (n) =  h(n) x((m − n)) N
n =0

m = 0,1,.......N − 1

where

x p (n) = x(n mod N ), −  n  

 Introduction

 Correlation is a mathematical operation that is

very similar to convolution. Just as with
convolution, correlation uses two signals to
produce a third signal. This third signal is called
the cross-correlation of the two input signals. If
a signal is correlated with itself, the resulting
signal is instead called the autocorrelation.

ES & BM Signal Processing 14

 Autocorrelation
◦ Correlating a signal with itself
 The problem

 Given a signal of some known shape, what is

the best way to determine where (or if) the
signal occurs in another signal..?? Correlation
is the answer.

ES & BM Signal Processing 16

 Correlation


rxy (l ) =  x ( n) y ( n − l )
n = −
l = 0,1,2,...

rxy (l ) =  x ( n + l ) y ( n)
n = −
l = 0,1,2,...

Where, rxy(l) is the correlation coefficients

 Autocorrelation can be used to extract a signal
from noise
 Cross correlation to locate signal
 Cross correlation can be used to detect and
locate known reference signal in noise
 Convolution vs. correlation

❑ Convolution is the relationship between a system's input

signal, output signal, and impulse response.
❑ Correlation is a way to detect a known waveform in a noisy
background.
❑ The similar mathematics is only a convenient coincidence.

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y(n)=-a1y(n-1)+b0x(n)+b1x(n-1)

v(n)=b0x(n)+b1x(n-1)

y(n)=-a1y(n-1)+v(n)

INTRODUCTION TO DIGITAL SPEECH PROCESSING (ET60007) © CET, IIT KGP

 Direct Form I
 Transfer function of recursive LTI system
N M
yn = − ak yn − k  +  bk xn − k 
k =1 k =0

M
vn =  bk xn − k  N
k =0 wn = − ak wn − k  + x(n)
N k =1

yn =  ak yn − k  + vn M

k =1 yn =  bk wn − k 
k =0

Copyright (C) 2005 Güner

351M Digital Signal Processing Arslan 22
Direct Form I

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 Copyright (C) 2005 Güner
351M Digital Signal Processing Arslan 24
Frequency-Domain Representation of Discrete
Signals and LTI Systems

x(n) = e jn LTI system y ( n)

complex-valued h( n) LTI system output
exponencial signal

impulse response


y ( n) =  h( k ) x ( n − k )
k =−

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LTI system output:

 
y ( n) =  h( k ) x ( n − k ) =
k =−
 h (
k =−
k ) e j ( n − k )
=
 
=  h (
k =−
k ) e − j k j n
e = e j n
 h (
k =−
k ) e − j k

y (n) = e jn H (e j )


Frequency response: H (e j ) =  h
k =−
( k ) e − j k

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H (e j ) = H (e j ) e j ( )

H (e j ) = Re  H (e j )  + j Im  H (e j ) 

 

H (e ) =  h(k )cos k + j  −  h(k )sin k 
j

k =−  k =− 

Re  H (e j )  =  h(k )cosk
k =−

Im  H (e j )  = −  h(k )sin k
k =−

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 Magnitude response:

j j 2 j 2
H (e ) = Re  H (e )  + Im  H (e ) 

Phase response:

Im  H (e j ) 
 ( ) = arg  H (e j )  = arctg j

Re  H (e ) 
Group delay function:

d ( )
 ( ) = −
d

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 Comments on relationship between the impulse response and
frequency response
The important property of the frequency response
 
H (e j ) = 
k =−
h ( k ) e − j k
=  h ( k )e
k =−
− j + 2 l 
= H (e
j + 2 l 
)

is fact that this function is periodic with period 2  .

In fact, we may view the previous expression as the exponential

j
Fourier series expansion for H (e ) , with h(k) as the Fourier series
coefficients. Consequently, the unit impulse response h(k) is related
j
to H (e ) through the integral expression

1
h( n) =
2 −
 H (e j  )e j  n d

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 Comments on symmetry properties
For LTI systems with real-valued impulse response, the magnitude response,
phase responses, the real component of and the imaginary component of
H (e j ) possess these symmetry properties:

The real component: even function of  periodic with period 2 

Re  H (e − j )  = Re  H (e j ) 
The imaginary component: odd function of  periodic with period 2

Im  H (e − j )  = − Im  H (e j ) 

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The magnitude response: even function of  periodic with period 2  2
H (e j ) = H (e − j )

The phase response: odd function of  periodic with period 2

arg  H (e − j )  = − arg  H (e j ) 

Consequence:
H (j
e )  ( )
If we known and for 0    , we can describe these
functions ( i.e. also H (e j ) ) for all values of  .

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 Symmetry H (e j ) EVEN
Properties


−4 −3 −2 − 0  2 3 4

OD
 ( )
D


−4 −3 −2 − 0  2 3 4

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 Normalized Frequency
It is often desirable to express the frequency response of an LTI system
in terms of units of frequency that involve sampling interval T. In this
case, the expressions:


1
j
H (e ) =  h ( k )e
k =−
− j k
h( n) =
2  H (e j  )e j  n d
−
are modified to the form:


H (e j T
)=  h(kT )e
k =−
− j kT

 /T
T
h(nT ) =
2 
− /T
H (e jT )e jnT d

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 j T is periodic with period 2  / T = 2 F, where F is
H ( e )
sampling frequency.
Solution: normalized frequency approach: F /2 → 

Example:

F = 100 kHz F / 2 = 50 kHz 50kHz → 

20 x103 2
f1 = 20 kHz 1 = 3
= = 0.4 
50 x10 5
25 x103 
f 2 = 25kHz 2 = 3
 = = 0.5 
50 x10 2

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 Z -Transform
Definition: The Z – transform of a discrete-time signal x(n) is
defined as the power series:

X ( z ) = Z [ x(n)]
X ( z) =  x (n) z
k =−
−k

where z is a complex variable. The above given relations

are sometimes called the direct Z - transform because
they transform the time-domain signal x(n) into its complex-
plane representation X(z).
Since Z – transform is an infinite power series, it exists only
for those values of z for which this series converges. The
region of convergence of X(z) is the set of all values of z for
which X(z) attains a finite value.

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The procedure for transforming from z – domain to the time-
domain is called the inverse Z – transform. It can be shown
that the inverse Z – transform is given by
1
xn =  X ( z ) z n −1
x(n) = Z −1  X ( z )
j 2 c

where C denotes the closed contour in the region of

convergence of X(z) that encircles the origin.

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