(lecture Outline #6)
The Discrete Time Fourier Transform
(DTFT)
(DTFT) - continued
DTFT Properties
LTI Systems and the Frequency Response
Magnitude and Phase Response
Black box
=
important
1
Copyright 2005, S. K. Mitra, 2006-2010 R.Leahy
Important Properties
Shift Properties
y (n M )
e jM Y (e j )
y ( n ) e j 0 n
Y (e j 0 )
Proof:
2
Copyright 2005, S. K. Mitra, 2006-2010 R.Leahy
Important Properties
Convolution Theorems
y (k ) x(n k )
Y ( e j ) X ( e j )
k =
y ( n) x ( n)
1
2
Y (e j ) X (e j )d
Proof:
3
Copyright 2005, S. K. Mitra, 2006-2010 R.Leahy
Important Properties
Differentiation
nx(n)
dX (e j )
j
d
Proof:
4
Copyright 2005, S. K. Mitra, 2006-2010 R.Leahy
Parsevals Theorem
x ( n) y ( n) =
n =
*
1
2
X (e j )Y (e j )d
Proof:
5
Copyright 2005, S. K. Mitra (edited: R.Leahy 2006, 2008)
DTFT Properties
Example - Determine the DTFT Y (e j ) of
y[n] = (n + 1) n[n], < 1
x[n] = n[n], < 1
Let
We can therefore write
y[n] = n x[n] + x[n]
From Table 3.3, the DTFT of x[n] is given
th DTFT
by
1
X ( e j ) =
1 e j
6
Copyright 2005, S. K. Mitra, 2006-2010 R.Leahy
DTFT
DTFT Properties
Using the differentiation property of the
DTFT given in Table
DTFT given in Table 3.2, we observe that
we observe that
the DTFT of n x[n] is given by
dX (e j )
d
1
e j
j
=j
1 e j = (1 e j ) 2
d
d
Next using the linearity property of the
using the linearity property of the
DTFT given in Table 3.4 we arrive at
e j
1
1
+
=
Y (e ) =
(1 e j ) 2 1 e j (1 e j ) 2
j
7
Copyright 2005, S. K. Mitra, 2006-2010 R.Leahy
The
The Frequency Response
Consider the LTI discrete-time system with
an impulse response
an impulse response {h[n]} shown below
shown below
x[n]
h[n]
y[n]
Its input-output relationship in the timedomain is given by the convolution sum
is given by the convolution sum
y[n] =
h[k ] x[n k ]
k =
8
Copyright 2005, S. K. Mitra, 2006-2010 R.Leahy
The
The Frequency Response
If the input is of the form
x[n] = e j n, < n <
then it follows that the output is given by
y[n] = h[k ] e
Let
j( n k )
k =
= h[k ] e j k e j n
k =
H (e j ) = h[k ] e j k
k =
9
Copyright 2005, S. K. Mitra, 2006-2010 R.Leahy
The
The Frequency Response
Then we can write
we can write
y[n] = H (e j ) e j n
Thus for a complex exponential input signal
e j n , the output of an LTI discrete-time
system is also a complex exponential signal
of the same frequency multiplied by a
the
j
complex constant H (e )
j n
Thus e
is an eigenfunction
(or eigensequence) of the system
10
Copyright 2005, S. K. Mitra, 2006-2010 R.Leahy
The
The Frequency Response
The quantity H (e j ) is called the frequency
response of the LTI discrete-time system
the
discrete
system
H (e j ) provides a frequency-domain
description of the system
H (e j ) is precisely the DTFT of the impulse
response {h[n]} of the system
11
Copyright 2005, S. K. Mitra, 2006-2010 R.Leahy
The
The Frequency Response
H (e j ) can be expressed in terms of its
real and imaginary parts
real and imaginary parts
H (e j ) = H re (e j ) + j H im (e j )
or, in terms of its magnitude and phase
response:
H (e j ) = H (e j ) e j()
where
12
() = arg H (e j )
Copyright 2005, S. K. Mitra, 2006-2010 Frequency R.Leahy
The
The Response
In some cases, the magnitude function is
specified in decibels as
G ( ) = 20 log10 H (e j ) / H (e j 0 )
13
dB
where G() is called the gain.
Note that this is a ratio; here I defined it
relative to the gain at =0.
Be careful since gain of zero
Be careful since a gain of zero
corresponds to dBs. (In Matlab you
need to set a minimum value if using
autoscaling)
Attenuation = - Gain in dB
Copyright 2005, S. K. Mitra, 2006-2010 R.Leahy
The
The Frequency Response
Example - Consider the M-point moving
average filter with an impulse response
average filter with an impulse response
given by
h[n] = 1 / M , 0 n M 1
0,
otherwise
Its frequency response is then given by
H ( e j ) =
=
14
1
M
M 1
j n
e
n =0
1 sin( M / 2) j ( M 1) / 2
e
M sin( / 2)
Copyright 2005, S. K. Mitra, 2006-2010 R.Leahy
The
The Frequency Response
Thus, the magnitude response of the M-point
moving average filter is given by
H ( e j ) =
1
M
sin( M / 2)
sin( / 2)
and the phase response is given by
( M 1) M/2
2 k
+ ( M )
() =
2
k =0
15
note second term gives -discontinuties as gain
passes through zero. More on phase soon .
Copyright 2005, S. K. Mitra, 2006-2010 R.Leahy
The
The Frequency Response
Magnitude
12
10
Phase
3
M=11
2
8
1
6
0
4
-1
2
-2
0
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-3
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Magnitude and phase response of the moving average filter.
Note linear phase with -discontinuities.
16
Copyright 2005, S. K. Mitra, 2006-2010 R.Leahy
DTFT,
DTFT, LTI and Frequency
Response
Consider the linear constant coefficient
difference equation:
difference equation:
N
M
k =0
k =0
d k y[n k ] = pk x[n k ]
Multiply each side by e jn and sum over n:
N
d
n = k = 0
k
y[n k ]e
j n
=
M
p x[n k ]e
n = k = 0
17
j n
k
Copyright 2005, S. K. Mitra, 2006-2010 R.Leahy
DTFT,
DTFT, LTI and Frequency
Response
Gives the relationship:
N
Y (e
j
) d k e
jk
= X (e
k =0
By the convolution theorem
y ( n) =
h( k ) x ( n k )
j
M
) p k e jk
k =0
Y (e j ) = H (e j )( X (e j )
k =
M
j
Y (e )
=
X ( e j )
18
p e
j k
d e
j k
k =0
N
k =0
k
= H ( e j )
k
Copyright 2005, S. K. Mitra, 2006-2010 R.Leahy
DTFT
DTFT Computation Using
MATLAB
The function freqz can be used to
compute the Frequency response of an LTI
compute the Frequency response of an LTI
system from the coefficients of the LDE:
p0 + p1e j + .... + pM e jM
H (e ) =
d 0 + d1e j + .... + d N e jN
j
at a prescribed set of discrete frequency
points =
19
Copyright 2005, S. K. Mitra, 2006-2010 R.Leahy
DTFT
DTFT Computation Using
MATLAB
For example, the statement
H = freqz(num,den,w)
returns the frequency response values as a
vector H of a DTFT defined in terms of the
vectors num and den containing the
coefficients { pi } and {di } , respectively at a
prescribed set of frequencies between 0 and
given by the vector w
20
Copyright 2005, S. K. Mitra, 2006-2010 R.Leahy
Example
A digital Chebyshev filter:
[b,a]=cheby1(5,.5,.5);
freqz(b,a);
21
Copyright 2005, S. K. Mitra, 2006-2010 R.Leahy
Example
A digital IIR Butterworth filter
[b,a]=butter(30,.5);
freqz(b,a);
22
Copyright 2005, S. K. Mitra, 2006-2010 R.Leahy