Tik-61.140 Signal Processing Systems
Spring 2001
Fourier Series Representation
of Continuous-Time
Periodic Signals
Fourier Series Representation
• Focus on the representation of continuoustime and discrete-time periodic signals
referred to as Fourier series
• Powerful and important tools for analyzing,
designing, and understanding signals and
LTI systems
Tik-61.140 / Chapter 3
 Olli Simula
The response of LTI systems to complex exponentials
Representation of signals as linear combinations of
basic signals that have the following properties:
1) The set of basic signals can be used to construct a broad and
useful class of signals
2
The response of LTI systems to complex exponentials
• The response of an LTI system to a complex
exponential is the same complex exponential with
only a change in amplitude:
– Continuous-time:
e sT -> H(s) e sT
– Discrete-time:
z n -> H(z) z n
where the complex amplitude factor H(s) or H(z) will
be a function of the complex variable s or z
2) The response of an LTI system to each signal should be
simple enough in structure to provide us with the convenient
representation for the response of the system to any signal
constructed as a linear combination of the basic signals
Both of these properties are provided by the set of
complex exponential signals in CT and DT
 Olli Simula
Tik-61.140 / Chapter 3
3
Tik-61.140 / Chapter 3
 Olli Simula
4
Continuous-Time Systems
The response of LTI systems to complex exponentials
• For an input x(t)=est the convolution integral gives:
• A signal for which the system output is a constant
times the input is referred to as an eigenfunction of the
system, and the amplitude value is referred as the
eigenvalue of the system
• This property for complex exponentials can be shown
using:
– The impulse response and
– The convolution
y (t ) =
=
+∞
+∞
−∞
−∞
∫ h (τ ) x(t −τ ) dτ = ∫ h(τ )e
+∞
∫ h (τ )e
st −sτ
e
dτ = est
−∞
+∞
+∞
s( t −τ )
∫ h(τ )e
dτ
−s τ
dτ
−∞
= x(t ) ∫ h (τ )e− sτ d τ
−∞
 Olli Simula
Chapter 3 / O. Simula
Tik-61.140 / Chapter 3
5
 Olli Simula
Tik-61.140 / Chapter 3
6
1
Tik-61.140 Signal Processing Systems
Spring 2001
Continuous-Time Systems
Discrete-Time Systems
• A complex exponential x(t)=est is now the eigenfunction of
the LTI system with impulseresponseh(t):
+∞
y(t ) = x(t ) ∫ h (τ )e
−∞
where
H ( s) =
− sτ
d τ = H (s )e
+∞
∫ h (τ )e
− sτ
• For an input x[n]=zn the convolution sum gives:
y [n ] =
st
+∞
+∞
k = −∞
k =−∞
∑ h[k ]x[n − k ] = ∑ h[k ]z n −k
y[n ] = z n H (z ) = x[ n]H ( z ) ,
dτ
where H ( z) =
−∞
Tik-61.140 / Chapter 3
+∞
∑ h[k ]z − k
k = −∞
+∞
∑ h[k ]z − k
k =−∞
H(s) is the transfer function or system function
representing the system behavior in the s-domain
 Olli Simula
= zn
H(z) is the z-transform of the unit impulse response
H(z) describes the system behavior in the z-domain
7
Tik-61.140 / Chapter 3
 Olli Simula
Linear Combination of Signals
8
Linear Combination of Signals
• From the superposition property the response to the sum is
the sum of the responses:
• Let x(t) correspond to a linear combination of complex
exponentials:
x ( t) = a1e s1 t + a 2e s2 t + a3e s 3t
• From the eigenfunction property the responseto each term:
a1e s1t → a1 H ( s1) e s1t
y (t ) = a1 H ( s1 ) e s1t + a2 H ( s2 ) e s 2t + a3 H ( s3 ) e s 3t
• The representation of signals as a linear combination of
complex exponentials leads to a convenient expression for
the response of an LTI system
a2e s2 t → a 2 H (s 2 )e s2 t
Input :
a3e s 3t → a3 H ( s3 ) e s3 t
Output :
x ( t) = ∑ a ke sk t
k
y ( t) = ∑ ak H ( sk )e sk t
k
Tik-61.140 / Chapter 3
 Olli Simula
9
 Olli Simula
Tik-61.140 / Chapter 3
10
Linear Combination of Signals
Fourier Series Representation
of Continuous
Continuous--Time
Periodic Signals
• In a similar way, for discrete-time systems the response of a
linear combination of complex exponentials is the linear
combination of individual responses :
Input :
x[n] = ∑ ak z k
n
k
Output :
y [n] = ∑ ak H ( zk ) zkn
k
The decompositionof more general signals in terms
of eigenfunctions is the basis for frequency domain
representation and analysis of LTI systems
 Olli Simula
Chapter 3 / O. Simula
Tik-61.140 / Chapter 3
11
 Olli Simula
2
Tik-61.140 Signal Processing Systems
Spring 2001
Linear Combinations of Harmonically
Related Complex Exponentials
Linear Combinations of Harmonically
Related Complex Exponentials
• A signal is periodic if for some value of T:
x (t) = x (t + T ) ,
for
• Basic periodic signals
all t
• The fundamental period of x(t) is the minimum positive,
nonzero value of T for which the above is satisfied;
• The value ω0=2π/T is referred to as the fundamental
frequency
Tik-61.140 / Chapter 3
 Olli Simula
13
x(t ) = e
jω 0 t
Tik-61.140 / Chapter 3
 Olli Simula
14
Linear Combination of Harmonically
Related Complex Exponentials
x( t ) =
φk (t) = e jk ω 0t = e jk (2π / T )t , k = 0, ±1, ±2,...
+∞
+∞
∑ ak e jk ω t = ∑ ak e jk (2π / T )t
0
k =−∞
• Each of these signals has a fundamental frequency that is a
multiple of ω 0
• Each is periodic with period T
• For |k|>2, the fundamental period of φk (t) is a fraction of T
Tik-61.140 / Chapter 3
15
k = −∞
• x(t) is periodic with period T
• The term for k=0 is a constant
• The terms for k=+1 and k=-1 both have fundamental
frequency equal to ω 0 and referred to as the fundamental
components or the first harmonic components
• The two terms for k=+2 and k=-2 are periodic with half the
period of the fundamental components and referred to as
the second harmonic components
Tik-61.140 / Chapter 3
 Olli Simula
Example 3.2
16
Fourier Series Representation
• Construction of the signal x(t) as linear
combination of harmonically related sinusoidal
signals
+3
x (t ) = ∑ a k e jk 2πt
• Periodic signal:
• In general, the components for k=+N and k=-N are also
periodic with a fraction of the period of the fundamental
components and are referred to as the Nth harmonic
components
k =−3
where a 0 = 1,
x( t) = cosω0t
– Complex exponential:
• Both of these signals are periodic with fundamental
frequency ω 0 and fundamental period of T=2π/ω0
Harmonically Related Complex
Exponentials
 Olli Simula
– Sinusoidal signal :
a1 = a −1 =
(
1
,
4
a2 = a− 2 =
1
2
) (
a3 = a −3 =
) (
1
3
x( t) = 1+
1 j2π t − j2π t 1 j4πt
1
e
+e
+ e
+ e − j4πt + e j6πt + e − j6πt
4
2
3
x( t) = 1+
1
2
cos(2πt ) + cos(4πt ) + cos(6π t)
2
3
 Olli Simula
Chapter 3 / O. Simula
Tik-61.140 / Chapter 3
x( t ) =
)
+∞
+∞
∑ ak e jk ω t = ∑ ak e jk (2π / T )t
0
k = −∞
k = −∞
=> Fourier Series representation
17
 Olli Simula
Tik-61.140 / Chapter 3
18
3
Tik-61.140 Signal Processing Systems
Spring 2001
Determination of the Fourier Series
Representation of a CT Periodid Signal
Determination of the Fourier Series
Representation of a CT Periodid Signal
T
∫ x( t)e
• Multiplying both sides and integrating gives:
+∞
x( t )e − jn ω 0t =
∑ ak e jk ω t e− jnω t
0
− jnω 0 t
∫ x( t)e
− jnω 0 t
∑ ak e jk ω t e − jn ω t dt
0
0
0 k = −∞
0
• The expression for determining the coefficients an is:
T

j ( k − n)ω 0 t 
= ∑ ak  ∫ e
dt


k = −∞  0
+∞
Tik-61.140 / Chapter 3
 Olli Simula
T
an =
+∞
Tik-61.140 / Chapter 3
 Olli Simula
• Let us approximate a given periodic signal x(t) by a linear
combination of a finite number of harmonically related
complex exponentials
+∞
0
k = −∞
1
1
− jk ω 0t
a k = ∫ x (t )e
dt = ∫ x (t ) e − jk ( 2π / T ) t dt
T T
T T
x N (t ) =
• The set of coefficients {ak} are called the Fourier series
coefficients or spectral coefficients of x(t)
• These complex coefficients measure the portion of the
signal x(t) that is at each harmonic of the fundamental
component
Tik-61.140 / Chapter 3
 Olli Simula
20
Convergence of the Fourier Series
∑ ak e jk ω t = ∑ ak e jk (2π / T )t
k = −∞
1
− jnω 0 t
x( t )e
dt
T∫
0
19
Fourier Series Representation
x (t ) =

T , k = n

j ( k − n)ω 0t
dt = 
∫e
 0, k ≠ n
0
T +∞
dt = ∫

0
T
k = −∞
T
T
∑ ak  ∫ e j (k− n )ω t dt 
k = −∞  0
0
0
+∞
dt =
+N
∑ ak e jkω t
0
k =−N
• Let eN(t) denote the approximation error
e N (t ) = x(t ) − xN (t ) = x (t ) −
21
+N
∑ ak e jk ω t
0
k =−N
Tik-61.140 / Chapter 3
 Olli Simula
Properties of the CT Fourier Series
Convergence of the Fourier Series
x ( t ) Periodic with period T and


y (t ) fundamenta l frequency ω 0 = 2π / T
• Quantitative measure for the goodness of the approximation
is defined by the energy in the error over one period
T
• It can be shown that the particular choice for coefficients
that minimize the energy in the error is
Aa k + Bb k
Time shifting :
x (t − t 0 )
ak e
Frequency shifting :
x (t ) e jM ω 0 t
ak − M
∫ x (τ ) y (t − τ ) dτ
Ta k bk
T
T
bk
Ax (t ) + By (t )
Periodic convolution :
1
− jk ω 0t
x (t )e
dt
T∫
ak
Linearity :
2
E N = ∫ e N (t ) dt
ak =
22
Multiplication :
x (t ) y( t )
− jk ω 0 t0
+∞
∑ al bk − l
l = −∞
 Olli Simula
Chapter 3 / O. Simula
Tik-61.140 / Chapter 3
23
 Olli Simula
Tik-61.140 / Chapter 3
24
4
Tik-61.140 Signal Processing Systems
Spring 2001
Linear Combinations of Harmonically
Related Complex Exponentials
Fourier Series Representation
of Discrete
Discrete--Time
Periodic Signals
• A discrete-time signal is periodic with period N if
x[ n] = x[ n + N ]
• The fundamental period is the smallest positive integer for
which the above equation holds
• ω 0=2π/N is the fundamental frequency
The Fourier series representation of a
discrete-time periodic signal is a finite
series, as opposed to the infinite series
representation required for continuous-time
periodic signals
 Olli Simula
Tik-61.140 / Chapter 3
 Olli Simula
Linear Combinations of Harmonically
Related Complex Exponentials
Linear Combinations of Harmonically
Related Complex Exponentials
• A set of all DT complex exponential signals that are periodic
with period N is given by
φk [ n] = e jk ω 0n = e jk ( 2π / N )n ,
26
k = 0, ±1, ± 2,...
• There are only N distinct signals in the above set due to the
fact that DT complex exponentials which differ in frequency
by a multiple of 2π are identical
φk [ n] = φ k + rN [n ]
• When k is changed by any integer multiple of N ,
the identical sequence is generated
• This differs from the situation in continuous-time in which
the signals φk (t) are all different from one another
φk [ n] = φ k + rN [n ]
Tik-61.140 / Chapter 3
 Olli Simula
27
k
k
• Multiplying both sides and summing over N terms:
k
x[ n]e − jr (2π / N )n =
• Since φk [n] are distinct only over a range on N successive values
of k, the summation need only include terms over this range
• Expressing the limits of summation as k = N
x[ n] =
∑ akφ k [n ] = ∑ a ke
k= N
jk ω 0n
k= N
=
∑ ak e
Chapter 3 / O. Simula
Tik-61.140 / Chapter 3
∑ x [n]e − jr ( 2π / N )n = ∑ ∑ a ke j (k −r )(2π / N )n
n= N
jk (2π / N )n
k= N
n= N k = N
=
29
∑ ak e j ( k −r )(2π / N )n
k= N
∑
ak
k= N
• This is referred to as the discrete-time Fourier series and the
coefficients ak as the Fourier series coefficients
 Olli Simula
28
Discrete-Time Fourier Series
Linear Combinations of Complex Exponentials
x[ n] = ∑ ak φ k [n ] = ∑ a k e jk ω 0n = ∑ ak e jk (2π / N ) n
Tik-61.140 / Chapter 3
 Olli Simula
 Olli Simula
Tik-61.140 / Chapter 3
∑ e j (k − r )(2π / N )n
n= N
30
5
Tik-61.140 Signal Processing Systems
Spring 2001
Discrete-Time Fourier Series
∑
∑
Discrete Fourier Series Representation
∑
x [n] e− jr ( 2π / N ) n =
ak
e j ( k − r )( 2π / N ) n
n= N
k= N
n= N
 N,

∑ e jk ( 2π / N ) n = 
• The synthesis and analysis equations for the discrete-time
Fourier series is given by the following pair of equations:
x[ n] =
k = 0 , ± N , ±2 N ,...
 0 , otherwise
n= N
1
∑ x[ n]e − jr (2π / N )n
N n= N
31
 Olli Simula
∑ akφ k [n ]
• If we take k in the range from 0 to N-1, we have
x[ n] = a0φ 0 [n ] + a1φ1[ n] + ...+ a N −1φ N − 1[n ]
• The values of Fourier coefficients ak repeat periodically
with period N
• Similarly, if k ranges from 1 to N, we have
x[ n] = a1φ1[ n] + a2φ 2 [n ] + ... + a N φ N [n ]
Since there are only N distinct complex exponentials that
are periodic with period N, the discrete-time Fourier series
representation is a finite series with N terms
• From φk [n]= φk+rN[n] we have φ0[n]= φN[n]
• Thus, we conclude that a0=aN
Tik-61.140 / Chapter 3
32
• Letting k range over any set of N consecutive integers we
conclude that
ak = ak + N
k= N
 Olli Simula
Tik-61.140 / Chapter 3
Discrete Fourier Series Representation
Discrete Fourier Series Representation
x[ n] =
k= N
• The set of coefficients {ak} are called thediscrete-time
Fourier series coefficients or spectral coefficients of x[n]
Tik-61.140 / Chapter 3
 Olli Simula
0
1
1
ak =
∑ x[ n]e − jkω0 n = N ∑ x[ n]e − jk (2π / N) n
N n= N
n= N
• Now, the expression for determining the coefficients an is:
an =
∑ ak e jk ω n = ∑ a ke jk (2π / N )n
k= N
33
 Olli Simula
Tik-61.140 / Chapter 3
34
Properties of the DT Fourier Series
x [n]  Periodic with period N and


y[n ] fundamental frequency ω 0 = 2π / N
ak  Periodic with


bk  period N
Linearity :
Ax[n ] + By[n ]
Aa k + Bb k
Time shifting :
x[n − n 0 ]
a k e− jk ( 2π / N ) n0
Frequency shifting :
x[n ]e
Periodic convolution :
jM ( 2π / N ) n
∑ x[ r] y[ n − r ]
r= N
Multiplication :
x[n ] y[ n ]
Fourier Series and
Linear Time
Time--Invariant
Systems
ak − M
Na k bk
∑ al bk − l
r= N
 Olli Simula
Chapter 3 / O. Simula
Tik-61.140 / Chapter 3
35
6
Tik-61.140 Signal Processing Systems
Spring 2001
Response of an LTI System
Fourier Series and LTI Systems
In continuous-time:
• Fourier series representation can be used to
construct any periodic signal in discrete-time
and essentially all periodic continuous-time
signals of practical importance
• The response of an LTI system to a linear
combination of complex exponentials take a
simple form
Tik-61.140 / Chapter 3
 Olli Simula
37
x( t ) = est ;
−∞
x[n ] = z n ;
 Olli Simula
∑
39
(
 Olli Simula
Chapter 3 / O. Simula
Tik-61.140 / Chapter 3
+∞
∑ h[n ]e− jωn
Tik-61.140 / Chapter 3
40
)
Need to change the relative amplitudes of
the frequency components in a signal or
eliminate some frequency components
entirely
The effect of the LTI system is to modify individually each
of the Fourier coefficients of the input by multiplying it
with the value of the frequency response
 Olli Simula
38
Filtering
ak H e j 2π k / N e jk ( 2π / N ) n
k= N
k = −∞
H(z) is the frequency response of the system in
discrete-time
∑ ak e jk (2π / N )n
k= N
+∞
∑ h[k ]z − k
n =−∞
• The response of an LTI system to a complex exponential
signal of the form ejωt or ejωn is simple to express
in terms of the frequency response
y [n ] =
H ( z) =
Tik-61.140 / Chapter 3
H (e jω ) =
h (t )e− jωt dt
Frequency Response
x[n ] =
where
• In discrete-time, we focus on values of z for which |z|=1,
so that z= ejω and zn= ejωn
• The system function H(z) for z of the form z= ejω is given by
+∞
Tik-61.140 / Chapter 3
• In discrete-time:
y[ n] = H ( z ) z n
When s and z are general complex numbers,
H(s) and H(z) are referred to as system functions
H(jω) is the frequency response of the system in
continuous-time
 Olli Simula
H (s ) = ∫ h (τ )e −sτ dτ
Frequency Response
• With Re{s}=0, i.e., s=jω, and consequently the input
x(t)=est = ejωt is the complex exponential at frequency ω
• The system function H(jω ) as a function of ω is given by
−∞
where
In discrete-time:
Frequency Response
H ( j ω) = ∫
+∞
y(t ) = H (s )est
41
=> FILTERING PROCESS
 Olli Simula
7
Tik-61.140 Signal Processing Systems
Spring 2001
Filtering operations
Filtering operations
• Frequency-shaping filters are linear timeinvariant systems that change the shape of the
spectrum
• Frequency-selective filters are designed to pass
some frequencies and significantly attenuate or
eliminate others
• Fourier series coefficients of the output of an LTI
system are those of the input multiplied by the
frequency response of the system
Tik-61.140 / Chapter 3
 Olli Simula
• Filtering can be conveniently accomplished
through the use of LTI systems with an
appropriately chosen frequency response
• Frequency-domain methods provide us with the
ideal tools to examine this important class of
applications
• Examples:
– Frequency shaping filters: Equalizer structures
– Frequency selective filters: Differentiating filters
43
Example: Image Filtering
Tik-61.140 / Chapter 3
 Olli Simula
44
Example: Two-Point Averaging Filter
x[n]
x[ n − 1]
D
1
2
+
• Impulse response:
h[n] =
y[ n ] =
1
(x [n] + x[n − 1])
2
1
(δ[n] + δ[n −1])
2
• Frequency response:
H( e j ω ) =
[
]
1
1 + e − jω = e− jω / 2 cos(ω / 2)
2
(
)
H (e jω ) = cos(ω / 2), and arg H (e jω ) = −ω / 2
 Olli Simula
Tik-61.140 / Chapter 3
45
• |H(e jω)| is large for frequencies
near ω=0 and decreasesas ω
approaches πor -π
• Higher frequencies are
attenuated more than lower
ones
• The phase response is linear
• Discrete -time frequency
response is periodic with
period 2π
Chapter 3 / O. Simula
Tik-61.140 / Chapter 3
Tik-61.140 / Chapter 3
46
Example: Two-Point Averaging Filter
Example: Two-Point Averaging Filter
 Olli Simula
 Olli Simula
47
• Consider a constant input, i.e., a zero-frequency complex
exponential
x[n] = Kej0n = K
The output is
y[n] = H(ej0n)Kej0n =[e-j0/2cos(0/2)]Kej0n = K = x[n]
• If the input is the high-frequency signal
x[n] = Kejπn = K(-1)n
The output is
y[n] = H(eπn)Keπn = [e-jπ/2cos(π/2)]Keπn = 0
• The system separates out the long-term constant value of a
signal from its high-frequency fluctuations
 Olli Simula
Tik-61.140 / Chapter 3
48
8
Tik-61.140 Signal Processing Systems
Spring 2001
Frequency-Selective Filters
Frequency-Selective Filters
• Lowpass filter
• A class of filters specifically intended to accurately
or approximately select some bands
of frequencies and reject others
• Examples:
– Removing noisein certain bands
– Communication systems; channel separation
– Passes low frequencies, i.e., frequencies around ω=0 and attenuates
or rejects higher frequencies
• Highpass filter
– Passes high frequencies and attenuates or rejects lower frequencies
• Bandpass filter
– Passes a band of frequencies and attenuates or rejects frequencies
both higher and lower than those in the band that is passed
• Bandstop filter
– Attenuates or rejects a band of frequencies and passes frequencies
both higher and lower than those in the band that is rejected
Tik-61.140 / Chapter 3
 Olli Simula
49
Tik-61.140 / Chapter 3
 Olli Simula
Frequency-Selective Filters
50
Ideal Frequency-Selective Filters
• Ideal lowpass filter with cutoff frequency ω c
• Cutoff frequencies
– The frequencies defining the boundaries between frequencies
that are passed and frequencies that are rejected , i.e.,
frequencies in the passband and in the stopband
 1, | ω |≤ ωc

H( jω ) = 
 0, | ω |> ω c
• Notch filter
– A bandstop filter which rejects a specific frequency and passes all
other frequencies
H(jω)
1
• Multiband filter
– A filter that has several passbands and stopbands
−ωc
• Comb filter
Tik-61.140 / Chapter 3
 Olli Simula
51
• Ideal bandpass filter with cutoff frequencies ωc1 and ω c2
 0, | ω |< ωc

H( jω ) = 
 1, | ω |≥ ωc
1, ω c1 ≤| ω |≤ ωc 2

H( jω ) = 
0, elsewhere
H(jω)
H(jω)
1
−ωc
 Olli Simula
Chapter 3 / O. Simula
0
Stopband
52
Ideal Frequency-Selective Filters
• Ideal highpass filter with cutoff frequency ω c
Passband
ω
Stopband
Tik-61.140 / Chapter 3
 Olli Simula
Ideal Frequency-Selective Filters
ωc
0
Passband
Stopband
– A multiband filter in which passbands and/or stopbands are
(usually) equally spaced in frequency
1
ωc
Tik-61.140 / Chapter 3
−ωc2
ω
−ωc1
0
ωc1
ωc2
ω
Passband
53
 Olli Simula
Tik-61.140 / Chapter 3
54
9
Tik-61.140 Signal Processing Systems
Spring 2001
Ideal Frequency-Selective Filters
Ideal Frequency-Selective Filters
• Ideal bandstop filter with cutoff frequencies ωc1 and ω c2
• Each of these ideal continuous-time filters is symmetric
aboutω=0
• There are two passbands for the highpass and bandpass
filters and three passbands for the bandstop filter
0, ω c1 ≤| ω |≤ ωc 2

H( jω ) = 
1, elsewhere
H(jω)
• Ideal discrete-time filters frequency-selective filters are
defined in the similar way
• For discrete-time filters the frequency response is periodic
with period 2π, with
• Low frequencies near even multiples of π
• High frequencies near odd multiples of π
1
−ωc2
−ωc1
0
ωc1
ωc2
ω
Tik-61.140 / Chapter 3
 Olli Simula
55
Ideal Discrete-Time Frequency-Selective Filters
H(ejω)
• Lowpass
Tik-61.140 / Chapter 3
 Olli Simula
Ideal Discrete-Time Frequency-Selective Filters
H(ejω)
• Bandpass
1
-π
-2π
−ωc
1
ωc
0
π
2π ω
H(ejω)
• Highpass
-2π
-π −ωc2
−ωc1 0 ωc
-π
−ωc
2π ω
1
ωc
0
π
2π ω
Tik-61.140 / Chapter 3
 Olli Simula
ωc2 π
H(ejω)
• Bandstop
1
-2π
56
57
+ v r (t) -
First-Order RC Lowpass Filter
-2π
 Olli Simula
-π −ωc2
−ωc1 0 ωc
ωc2 π
2π ω
Tik-61.140 / Chapter 3
58
Transfer Function of the First -Order RC Lowpass Filter
R
RC
dvC ( t)
+ vC ( t) = v s ( t )
dt
Input voltage : v s ( t ) = e jω t ;
RC
[
v s(t) +
-
C
+
v (t)
- c
H ( jω ) =
Output voltage : vC ( t) = H ( jω) e jω t
]
• For frequencies near ω=0, |H(j ω )| is close to 1
d
H ( jω ) e jω t + H ( jω ) e jωt = e jω t
dt
• For larger values of ω, |H(j ω )| is considerably smaller
RCjω H ( jω ) e jω t + H ( j ω ) e jω t = e jω t
RCjω H ( jω ) + H ( jω ) = 1
 Olli Simula
Chapter 3 / O. Simula
1
1+ RCjω
Tik-61.140 / Chapter 3
• |H(j ω)| approaches zero when ω approaches infinity
H( jω ) =
1
1 + RCj ω
59
 Olli Simula
Tik-61.140 / Chapter 3
60
10
Tik-61.140 Signal Processing Systems
Spring 2001
Transfer Function of the First -Order RC Lowpass Filter
• The output is now the voltage
across the resistor
1 −t / RC
e
u (t )
RC
• Impulse response:
h(t ) =
• Step response:
s(t) = 1 − e
[
− t / RC
] u (t)
– Narrow passband requirement: Large RC
Tik-61.140 / Chapter 3
61
 Olli Simula
R
Input voltage : v s ( t ) = e jω t ;
RC
[
+
-
v s(t)
C
+ v (t)
- c
Output voltage : v R ( t) = G( j ω) e jω t
]
d
d
G( jω )e jω t + G ( jω )e jω t = RC e jω t
dt
dt
RCjω G( jω ) e jωt + G( jω ) e jω t = RCj ωe jωt
G( jω ) =
RCjω G ( jω ) + G ( j ω ) = RCjω
jω RC
1+ jω RC
Tik-61.140 / Chapter 3
 Olli Simula
63
+
Input : x [ n] = e jωn ;
Tik-61.140 / Chapter 3
62
• Discrete-time LTI systems described by difference
equations can be either
Recursive and have an infinite impulse response
(IIR systems)
or
Nonrecursive and have a finite impulse response
(FIR systems)
• IIR systems are direct counterparts of continuous-time
systems described by differential equations
 Olli Simula
Tik-61.140 / Chapter 3
64
y[n]
Difference equation:
y [ n ] − ay [ n − 1] = x [ n ]
dvR ( t)
dv ( t )
+ v R (t ) = RC s
dt
dt
Transfer Function of the First-Order Recursive
Discrete-Time Filter
First-Order Recursive Discrete-Time Filters
x [n]
+ v (t)
- c
Discrete-Time Filters Described by
Constant Coefficient Difference Equations
+ v r (t) -
dv R ( t)
dv ( t )
+ v R (t ) = RC s
dt
dt
C
dvC ( t )
d
= C (v s ( t ) − vR ( t ) )
dt
dt
RC
– Fast step response: Small RC
RC
+
-
 dv (t ) dv ( t ) 
v R ( t) = Ri ( t) = RC  s − R 
dt 
 dt
• Trade-offs in filter design:
First-Order RC Highpass Filter
R
v s(t)
v C (t ) = v s (t ) − v R (t )
i (t ) = C
 Olli Simula
+ v r (t) -
First-Order RC Highpass Filter
a
Output : y [n ] = H ( e jω ) e
H (e jω ) =
D
y[n − 1]
1
1− ae − jω
• Parameter a controls the behavior of the filter
jω n
• For a positive => Lowpass filter
H (e j ω )e j ωn − aH ( e jω ) e jω ( n −1) = e jωn
[1 − ae ] H (e
− jω
jω
 Olli Simula
Chapter 3 / O. Simula
)e jωn = e jω n
Tik-61.140 / Chapter 3
a controls the rate of attenuation at low frequencies, ω=0
H (e jω ) =
• For a negative => Highpass filter
1
1− ae − jω
65
a controls the rate of attenuation at high frequencies, ω=π
 Olli Simula
Tik-61.140 / Chapter 3
66
11
Tik-61.140 Signal Processing Systems
Spring 2001
Transfer Function of the First-Order Recursive
Discrete-Time Filter
Nonrecursive Discrete-Time Filters
• General form of an FIR nonrecursive difference equation
h[ n] = a n u[n]
• Impulse response:
n +1
1− a
s[ n] = u[ n] ∗ h[n ] =
1− a
• Step response:
u[ n]
With faster responses for smaller values of |a| , and
hence for broader passbands
• For |a|<1 the system is stable, i.e., h[n] is absolutely
summable
Tik-61.140 / Chapter 3
67
Three-Point Moving-Average Filter
y[ n ] =
jω
)=
h [n ] =
1
(δ [ n − 1] + δ [ n] + δ [ n + 1])
3
(
1 − jω
e
+ 1+ e
3
1
(δ [ n] + δ [ n − 1] + δ [ n − 2] )
3
• The frequency response
jω
)
=
H (e jω ) =
(
1
1 + e − j ω + e − j 2ω
3
1
= e − jω (1 + 2 cos ω )
3
1
(1 + 2 cos ω )
3
Tik-61.140 / Chapter 3
 Olli Simula
1
( x[ n ] + x [ n − 1] + x [n − 2] )
3
• The impulse response
• The frequency response
H (e
68
• Difference equation
1
(x [ n − 1] + x[ n ] + x [ n + 1] )
3
• The impulse response
h[ n] =
Tik-61.140 / Chapter 3
 Olli Simula
Causal Three-Point Moving-Average Filter
• Difference equation
y[ n ] =
k =−N
• The output is the weighted average of the (N+M+1) values
of x[n] from x[n-M] through x[n+N] with the weights given
by coefficients bk .
• Such a filter is often called a moving-average filter, where
the output y[n] for any n , e.g. for n0 , is an average of
values of x[n] in the vicinity of n0
• |a| controls the speed with which the impulse and
step responses approach their long-term values,
 Olli Simula
M
∑ bk x [ n − k ]
y[ n ] =
69
= e− jω
(
1 jω
e
+ 1 + e − jω
3
Tik-61.140 / Chapter 3
 Olli Simula
Three-Point Moving-Average Filter
)
)
70
General Moving-Average Filter
y[ n ] =
• Difference equation
M
1
x[n − k ]
∑
N + M + 1 k =− N
• The impulse response is a rectangular pulse, i.e.,
h[n]=1/(N+M+1) for -N<n<M and h[n]=0 otherwise
• The frequency response
H (e jω ) =
 Olli Simula
Chapter 3 / O. Simula
Tik-61.140 / Chapter 3
71
 Olli Simula
1
sin [ω ( N + M + 1) / 2 ]
e jω [( N − M ) / 2 ]
N + M +1
sin(ω / 2 )
Tik-61.140 / Chapter 3
72
12
Tik-61.140 Signal Processing Systems
Spring 2001
Moving-Average Filter with N+M+1=25
Tik-61.140 / Chapter 3
 Olli Simula
Moving-Average Filter with N+M+1=50
73
Differentiating Nonrecursive Filter
 Olli Simula
Tik-61.140 / Chapter 3
74
First-Order Nonrecursive Highpass Filter
• Consider the difference equation
y[ n ] =
(x [ n ] − x [ n − 1])
2
• For input signals that vary greatly from sample to sample,
the value of y[n] is large
1
• The impulse response
h[ n] = (δ [ n ] − δ [ n − 1])
2
• The frequency response
(
1
H ( e j ω ) = 1 − e − jω
2
) = je − j ω / 2 sin( ω / 2 )
Tik-61.140 / Chapter 3
 Olli Simula
75
 Olli Simula
Tik-61.140 / Chapter 3
76
Nonrecursive Discrete-Time Filters
• The impulse response of a nonrecursive FIR filter is
of finite length
• The impulse response is, thus, always absolutely
summable for any h[n]=bn
=> FIR filters are always stable
 Olli Simula
Chapter 3 / O. Simula
Tik-61.140 / Chapter 3
77
13