1. Introduction to Analog And Digital
Communications
Second Edition
Simon Haykin, Michael Moher
2. Chapter 2 Fourier Representation of Signals
and Systems
2.1 The Fourier Transform
2.2 Properties of the Fourier Transform
2.3 The Inverse Relationship Between Time and Frequency
2.4 Dirac Delta Function
2.5 Fourier Transforms of Periodic Signals
2.6 Transmission of Signals Through Linear Systems
: Convolution Revisited
2.7 Ideal Low-pass Filters
2.8 Correlation and Spectral Density : Energy Signals
2.9 Power Spectral Density
2.10 Numerical Computation of the Fourier Transform
2.11 Theme Example
: Twisted Pairs for Telephony
2.12 Summary and Discussion
3. 2.1 The Fourier Transform
Definitions
Fourier Transform of the signal g(t)
∞
G( f ) = ∫
−∞
g (t ) exp(− j 2πft )dt (2.1)
exp(− j 2πft ) : the kernel of thr formula defining the Fourier transform
∞
g (t ) = ∫
−∞
G ( f ) exp( j 2πft )df (2.2)
exp( j 2πft ) : the kernel of thr formula defining the Inverse Fourier transfom
A lowercase letter to denote the time function and an uppercase letter to denote the
corresponding frequency function
Basic advantage of transforming : resolution into eternal sinusoids presents the
behavior as the superposition of steady-state effects
Eq.(2.2) is synthesis equation : we can reconstruct the original time-domain
behavior of the system without any loss of information.
3
4. Dirichlet’s conditions
1. The function g(t) is single-valued, with a finite number of maxima and minima
in any finite time interval.
2. The function g(t) has a finite number of discontinuities in any finite time interval.
3. The function g(t) is absolutely integrable
∞
∫−∞
g (t ) dt < ∞
For physical realizability of a signal g(t), the energy of the signal defined by
∞
∫
2
g (t ) dt
−∞
must satisfy the condition
∞
∫ g (t ) dt < ∞
2
−∞
Such a signal is referred to as an energy signal.
All energy signals are Fourier transformable.
4
5. Notations
The frequency f is related to the angular frequency w as
w = 2πf [rad / s ]
Shorthand notation for the transform relations
G ( f ) = F[ g (t )] (2.3) g (t ) = F −1[G ( f )] (2.4)
g (t ) ⇔ G ( f ) (2.5)
5
6. Continuous Spectrum
A pulse signal g(t) of finite energy is expressed as a continuous sum of
exponential functions with frequencies in the interval -∞ to ∞.
We may express the function g(t) in terms of the continuous sum infinitesimal
components, ∞
g (t ) = ∫
−∞
G ( f ) exp( j 2πft )df
The signal in terms of its time-domain representation by specifying the function
g(t) at each instant of time t.
The signal is uniquely defined by either representation.
The Fourier transform G(f) is a complex function of frequency f,
G ( f ) = G ( f ) exp[ jθ ( f )] (2.6)
G ( f ) : continuous amplitude spectrum of g(t)
θ ( f ) : continuous phase spectrum of g(t)
6
7. For the special case of a real-valued function g(t)
G (− f ) = G * ( f )
G (− f ) = G ( f )
θ (− f ) = −θ ( f )
* : complex conjugation
The spectrum of a real-valued signal
1. The amplitude spectrum of the signal is an even function of the frequency,
the amplitude spectrum is symmetric with respect to the origin f=0.
2. The phase spectrum of the signal is an odd function of the frequency, the
phase spectrum is antisymmetric with respect to the origin f=0.
7
16. 2.2 Properties of the Fourier Transrom
Property 1 : Linearity (Superposition)
Let g1 (t ) ⇔ G1 ( f ) and g 2 (t ) ⇔ G2 ( f )
then for all constants c1 and c2,
c1 g1 (t ) + c2 g 2 (t ) ⇔ c1G1 ( f ) + c2G2 ( f ) (2.14)
Property 2 : Dialation
Let g (t ) ⇔ G ( f )
1 f
g (at ) ⇔ G (2.20)
a a
The dilation factor (a) is real number
∞ 1 ∞
f
F [ g (at )] = ∫
−∞
g (at ) exp(− j 2πft )dt F [ g (at )] =
−∞
a ∫
g (τ ) exp − j 2π τ dτ
a
1 f
= G
a a
16
17. The compression of a function g(t) in the time domain is equivalent to the
expansion of ite Fourier transform G(f) in the frequency domain by the same
factor, or vice versa.
Reflection property
For the special case when a=-1
g (−t ) ⇔ G (− f ) (2.21)
17
22. Property 3 : Conjugation Rule
Let g (t ) ⇔ G ( f )
then for a complex-valued time function g(t),
g * (t ) ⇔ G * (− f ) (2.22)
Prove this ;
∞
g (t ) = ∫
−∞
G ( f ) exp( j 2πft )df
∞
g (t ) = ∫ G * ( f ) exp(− j 2πft )df
*
−∞
∞
g (t ) = − ∫ G * (− f ) exp( j 2πft )df
*
−∞
∞
= ∫
−∞
G * (− f ) exp( j 2πft )df
g * (−t ) ⇔ G * ( f ) (2.23)
22
23. Property 4 : Duality
If g (t ) ⇔ G ( f )
G (t ) ⇔ g (− f ) (2.24)
∞
g (−t ) = ∫−∞
G ( f ) exp(− j 2πft )df
Which is the expanded part of Eq.(2.24) in going from the time domain to the
frequency domain.
∞
g (− f ) = ∫ −∞
G (t ) exp(− j 2πft )dt
23
26. Property 5 : Time Shifting
If g (t ) ⇔ G ( f )
g (t − t0 ) ⇔ G ( f ) exp(− j 2πft0 ) (2.26)
If a function g(t) is shifted along the time axis by an amount t0, the effect is
equivalent to multiplying its Fourier transform G(f) by the factor exp(-j2πft0).
Prove this ;
∞
F [ g (t − t0 )] = exp(− j 2πft0 ) ∫
−∞
g (τ ) exp(− j 2πτ )dτ
= exp(− j 2πft0 )G ( f )
Property 6 : Frequency Shifting
If g (t ) ⇔ G ( f )
exp( j 2πf c t ) g (t ) ⇔ G ( f − f c ) (2.27)
26
27. This property is a special case of the modulation theorem
A shift of the range of frequencies in a signal is accomplished by using the process
of modulation.
∞
F [exp( j 2πf c t ) g (t )] = ∫ g (t ) exp[− j 2πt ( f − f c )]dt
−∞
= G( f − fc )
Property 7 : Area Under g(t)
If g (t ) ⇔ G ( f )
∞
∫
−∞
g (t )dt = G (0) (2.31)
The area under a function g(t) is equal to the value of its Fourier transform G(f)
at f=0.
27
32. Property 8 : Area under G(f)
If g (t ) ⇔ G ( f )
∞
g ( 0) = ∫
−∞
G ( f )df (2.32)
The value of a function g(t) at t=0 is equal to the area under its Fourier
transform G(f).
Property 9 : Differentiation in the Time Domain
Let g (t ) ⇔ G ( f )
d
g (t ) ⇔ j 2πfG ( f ) (2.33)
dt
Differentiation of a time function g(t) has the effect of multiplying its Fourier
transform G(f) by the purely imaginary factor j2πf.
dn
g (t ) ⇔ ( j 2πf ) n G ( f ) (2.34)
dt n
32
36. Property 10 : Integration in the Time Domain
Let g (t ) ⇔ G ( f ) , G(0) = 0,
1
∫
t
g (τ )dτ ⇔ G (t ) (2.41)
−∞ j 2πf
Integration of a time function g(t) has the effect of dividing its Fourier
transform G(f) by the factor j2πf, provided that G(0) is zero.
verify this ;
g (t ) = g (τ )dτ
d
∫
t
dt
−∞
F t g (τ )dτ
G ( f ) = ( j 2πf ) ∫
−∞
36
41. Property 11 : Modulation Theorem
Let g1 (t ) ⇔ G1 ( f ) and g 2 (t ) ⇔ G2 ( f )
∞
g1 (t ) g 2 (t ) ⇔ ∫ −∞
G1 (λ )G2 ( f − λ )dλ (2.49)
We first denote the Fourier transform of the product g1(t)g2(t) by G12(f)
g1 (t ) g 2 (t ) ⇔ G12 ( f )
∞
G12 ( f ) =∫ −∞
g1 (t )g 2 (t ) exp(− j 2πft )dt
∞
g (t ) = ∫
2 G2 ( f ' ) exp(− j 2πf 't )df '
−∞
∞ ∞
G12 ( f ) = ∫ ∫ g1 (t )G2 ( f ' ) exp[− j 2π ( f − f ' )t ]df ' dt
−∞ −∞
G12 ( f ) = G2 ( f − λ ) g1 (t ) exp(− j 2πλt )dt dλ
∞ ∞
−∞∫
∫−∞
the inner integral is recognized simply as G1 (λ )
41
42. ∞
G12 ( f ) = ∫
−∞
G1 (λ )G2 ( f − λ )dλ
This integral is known as the convolution integral
Modulation theorem
The multiplication of two signals in the time domain is transformed into the
convolution of their individual Fourier transforms in the frequency domain.
Shorthand notation
G12 ( f ) = G1 ( f ) ∗ G2 ( f )
g1 (t ) g 2 (t ) ⇔ G1 ( f ) ∗ G2 ( f ) (2.50)
G1 ( f ) ∗ G2 ( f ) = G2 ( f ) ∗ G1 ( f )
42
43. Property 12 : Convolution Theorem
Let g1 (t ) ⇔ G1 ( f ) and g 2 (t ) ⇔ G2 ( f )
∞
∫−∞
g1 (τ ) g 2 (t − τ )dτ ⇔G1 ( f )G2 ( f ) (2.51)
Convolution of two signals in the time domain is transformed into the
multiplication of their individual Fourier transforms in the frequency domain.
g1 (t ) ∗ g 2 (t ) = G1 ( f )G2 ( f ) (2.52)
Property 13 : Correlation Theorem
Let g1 (t ) ⇔ G1 ( f ) and g 2 (t ) ⇔ G2 ( f )
∞
∫ g1 (t ) g 2 (t − τ )dt ⇔G1 ( f )G2* ( f ) (2.53)
*
−∞
The integral on the left-hand side of Eq.(2.53) defines a measure of the
similarity that may exist between a pair of complex-valued signals
∞
∫−∞
g1 (t ) g 2 (τ − t )dt ⇔G1 ( f )G2 ( f ) (2.54)
43
44. Property 14 : Rayleigh’s Energy Theorem
Let g (t ) ⇔ G ( f )
∞ ∞
∫ g (t ) dt = ∫
2 2
G ( f ) df (2.55)
−∞ −∞
Total energy of a Fourier-transformable signal equals the total area under the
curve of squared amplitude spectrum of this signal.
∞
∫ g (t ) g * (t − τ )dt ⇔G ( f )G ( f ) = G ( f )
* 2
−∞
∞ ∞ 2
∫ g (t ) g (t − τ )dt = ∫ G ( f ) exp( j 2πfτ )df
*
(2.56)
−∞ −∞
44
46. 2.3 The Inverse Relationship Between Time and Frequency
1. If the time-domain description of a signal is changed, the
frequency-domain description of the signal is changed in an inverse
manner, and vice versa.
2. If a signal is strictly limited in frequency, the time-domain
description of the signal will trail on indefinitely, even though its
amplitude may assume a progressively smaller value. – a signal
cannot be strictly limited in both time and frequency.
Bandwidth
A measure of extent of the significant spectral content of the signal for
positive frequencies.
46
47. Commonly used three definitions
1. Null-to-null bandwidth
When the spectrum of a signal is symmetric with a main lobe bounded by
well-defined nulls – we may use the main lobe as the basis for defining the
bandwidth of the signal
2. 3-dB bandwidth
Low-pass type : The separation between zero frequency and the positive
frequency at which the amplitude spectrum drops to 1/√2 of its peak value.
Band-pass type : the separation between the two frequencies at which the
amplitude spectrum of the signal drops to 1/√2 of the peak value at fc.
3. Root mean-square (rms) bandwidth
The square root of the second moment of a properly normalized form of the
squared amplitude spectrum of the signal about a suitably chosen point.
47
48. The rms bandwidth of a low-pass signal g(g) with Fourier transform G(f) as follows :
1/ 2
f G ( f ) df
∞
∫
2 2
Wrms = −∞ ∞ (2.58)
∫ G ( f ) df
2
It lends itself more readily − ∞ mathematical
to evaluation than the other two definitions
of bandwidth
Although it is not as easily measured in the lab.
Time-Bandwidth Product
The produce of the signal’s duration and its bandwidth is always a constant
(duration) × (bandwidth) = constant
Whatever definition we use for the bandwidth of a signal, the time-bandwidth
product remains constant over certain classes of pulse signals
48
49. Consider the Eq.(2.58), the corresponding definition for the rms duration of the
signal g(t) is
1/ 2
∫ t g (t ) dt
∞ 2
2
Trms = −∞
∞ (2.59)
∫ g (t ) dt
2
−∞
The time-bandwidth product has the following form
1
TrmsWrms ≥ (2.60)
4π
49
50. 2.4 Dirac Delta Function
The theory of the Fourier transform is applicable only to time functions
that satisfy the Dirichlet conditions
∞
∫ g (t ) dt < ∞
2
−∞
1. To combine the theory of Fourier series and Fourier transform into a unified
framework, so that the Fourier series may be treated as a special case of the
Fourier transform
2. To expand applicability of the Fourier transform to include power signals-that
is, signals for which the condition holds.
1 T
∫ g (t ) dt < ∞
2
lim 2T
T →∞
−T
Dirac delta function
Having zero amplitude everywhere except at t=0, where it is infinitely large in
such a way that it contains unit area under its curve.
50
51. δ (t ) = 0, t ≠ 0 (2.61)
∞
∫−∞
δ (t )dt = 1 (2.62)
We may express the integral of the product g(t)δ(t-t0) with respect to time t as
follows : ∞
∫
−∞
g (t )δ (t − t0 )dt = g (t0 ) (2.63)
Sifting property of the delta function
∞
∫−∞
g (τ )δ (t − τ )dτ = g (t ) (2.64)
g (t ) ∗ δ (t ) = g (t )
The convolution of any time function g(t) with the delta function δ(t) leaves
that function completely unchanged. – replication property of delta function.
The Fourier transform of the delta function is
∞
F [δ (t )] = ∫ −∞
δ (t ) exp(− j 2πft )dt
51
52. The Fourier transform pair for the Delta function
F [δ (t )] = 1
δ (t ) ⇔ 1 (2.65)
The delta function as the limiting form of a pulse of unit area as the
duration of the pulse approaches zero.
A rather intuitive treatment of the function along the lines described
herein often gives the correct answer.
Fig. 2.12
52
57. Applications of the Delta Function
1. Dc signal
By applying the duality property to the Fourier transform pair of Eq.(2.65)
1 ⇔ δ ( f ) (2.67)
A dc signal is transformed in the frequency domain into a delta function occurring at
zero frequency
∞
∫
−∞
exp(− j 2πft )dt = δ ( f )
∞
∫−∞
cos(2πft )dt = δ ( f ) (2.68)
2. Complex Exponential Function
By applying the frequency-shifting property to Eq. (2.67)
exp( j 2πf c t ) ⇔ δ ( f − f c ) (2.69)
Fig. 2.14
57
59. 3. Sinusoidal Functions
The Fourier transform of the cosine function
1
cos(2πf c t ) = [exp( j 2πf c t ) + exp(− j 2πf c t )] (2.70)
2
1
cos(2πf c t ) ⇔ [δ ( f − f c ) + δ ( f + f c )] (2.71)
2
Fig. 2.15
The spectrum of the cosine function consists of a pair of delta functions occurring at
f=±fc, each of which is weighted by the factor ½
1
sin( 2πf c t ) ⇔ [δ ( f − f c ) − δ ( f + f c )] (2.72)
2j
4. Signum Function + 1, t > 0
sgn(t ) = 0, t = 0
− 1, t < 0
Fig. 2.16
59
62. This signum function does not satisfy the Dirichelt conditions and therefore, strictly
speaking, it does not have a Fourier transform
exp(− at ), t >0
g (t ) = 0, t = 0 (2.73)
− exp(− at ), t < 0
Its Fourier transform was derived in Example 2.3; the result is given by
− j 4πf
G( f ) =
a 2 + (2πf ) 2
The amplitude spectrum |G(f)| is shown as the dashed curve in Fig. 2.17(b). In the
limit as a approaches zero,
− j 4πf
F[sgn(t )] = lim
a →0
a 2 + (2πf ) 2 sgn(t ) ⇔
1
(2.74)
1 jπf
=
jπf
At the origin, the spectrum of the approximating function g(t) is zero for a>0,
whereas the spectrum of the signum function goes to infinity.
Fig. 2.17
62
64. 5. Unit Step Function
The unit step function u(t) equals +1 for positive time and zero for negative time.
1, t > 0
1
u (t ) = , t = 0
2
0, t < 0
The unit step function and signum function are related by
1
u (t ) = [sgn((t ) + 1)] (2.75)
2
Unit step function is represented by the Fourier-transform pair
• The spectrum of the unit step function contains a delta function weighted by a factor of ½
and occurring at zero frequency
1 1
u (t ) ⇔ + δ ( f ) (2.76)
j 2πf 2
Fig. 2.18
64
66. 6. Integration in the time Domain (Revisited)
The effect of integration on the Fourier transform of a signal g(t), assuming that G(0)
is zero.
∫
t
y (t ) = g (τ )dτ (2.77)
−∞
The integrated signal y(t) can be viewed as the convolution of the original signal g(t)
and the unit step function u(t)
∞
y (t ) = ∫
−∞
g (τ )u (t − τ )dτ
1, τ < t
1
u (t − τ ) = , τ = t
2
0, τ > t
The Fourier transform of y(t) is
1 1
Y ( f ) = G( f ) + δ ( f ) (2.78)
j 2πf 2
66
67. The effect of integrating the signal g(t) is
G ( f )δ ( f ) = G (0)δ ( f )
1 1
Y( f ) = G ( f ) + G (0)δ ( f )
j 2πf 2
1 1
∫
t
g (τ )dτ ⇔ G ( f ) + G (0)δ ( f ) (2.79)
−∞ j 2πf 2
67
68. 2.5 Fourier Transform of Periodic Signals
A periodic signal can be represented as a sum of complex exponentials
Fourier transforms can be defined for complex exponentials
Consider a periodic signal gT0(t)
∞
gT0 (t ) = ∑ c exp( j 2πnf t )
n = −∞
n 0 (2.80)
1 T /2
cn = ∫ gT0 (t ) exp(− j 2πnf 0t )dt (2.81)
0
T0 −T / 2 0
Complex Fourier coefficient
1
f0 = (2.82)
T0
f0 : fundamental frequency
68
69. Let g(t) be a pulselike function
T0 T0
gT (t ), − ≤ t ≤
g (t ) = 0 2 2 (2.83)
0,
elsewhere
∞
gT0 (t ) = ∑ g (t − mT )
m = −∞
0 (2.84)
∞
cn = f 0 ∫
−∞
g (t ) exp(− j 2πnf 0t )dt
= f 0G (nf 0 ) (2.85)
∞
gT0 (t ) = f 0 ∑ G (nf 0 ) exp( j 2πnf 0t ) (2.86)
n = −∞
69
70. One form of Possisson’s sum formula and Fourier-transform pair
∞ ∞
∑ g (t − mT ) = f ∑ G(nf ) exp( j 2πnf t )
m = −∞
0 0
n = −∞
0 0
(2.87)
∞ ∞
∑ g (t − mT ) ⇔ f ∑ G(nf )δ ( f − nf )
m = −∞
0 0
n = −∞
0 0 (2.88)
Fourier transform of a periodic signal consists of delta functions occurring at
integer multiples of the fundamental frequency f0 and that each delta function
is weighted by a factor equal to the corresponding value of G(nf0).
Periodicity in the time domain has the effect of changing the
spectrum of a pulse-like signal into a discrete form defined at
integer multiples of the fundamental frequency, and vice versa.
70
74. 2.6 Transmission of Signal Through Linear Systems
: Convolution Revisited
In a linear system,
The response of a linear system to a number of excitations applied
simultaneously is equal to the sum of the responses of the system when each
excitation is applied individually.
Time Response
Impulse response
The response of the system to a unit impulse or delta function applied to the input
of the system.
Summing the various infinitesimal responses due to the various input pulses,
Convolution integral
The present value of the response of a linear time-invariant system is a weighted
integral over the past history of the input signal, weighted according to the
impulse response of the system
∞
y (t ) = ∫ −∞
x(τ )h(t − τ )dτ (2.93)
∞
y (t ) = ∫ h(τ )x(t − τ )dτ (2.94)
−∞
74
79. Causality and Stability
Causality
It does not respond before the excitation is applied
h(t ) = 0, t < 0 (2.98)
Stability
The output signal is bounded for all bounded input signals (BIBO)
x(t ) < M for all t
∞
y (t ) = ∫−∞
h(τ )x(t − τ )dτ (2.99)
Absolute value of an integral is bounded by the integral of the absolute value of the
integrand
∞ ∞
∫ h(τ )x(t − τ ) dτ ≤ ∫ h(τ ) x(t − τ ) dτ ∞
∫
−∞ −∞
∞
y(t) ≤ M h(τ ) dτ
= M ∫ h(τ ) dτ
−∞
−∞
79
80. A linear time-invariant system to be stable
The impulse response h(t) must be absolutely integrable
The necessary and sufficient condition for BIBO stability of a linear time-invariant
system
∞
∫−∞
h(t ) dt < ∞ (2.100)
Frequency Response
Impulse response of linear time-invariant system h(t),
Input and output signal
x(t ) = exp( j 2πft ) (2.101)
∞
y (t ) = ∫ h(τ ) exp[ j 2πf (t − τ )]dτ
−∞
∞
= exp( j 2πft ) ∫ h(τ ) exp(− j 2πfτ )dτ (2.102)
−∞
80
81. ∞
H( f ) = ∫ h(t ) exp(− j 2πft )dt (2.103) y (t ) = H ( f ) exp( j 2πft ) (2.104)
−∞
Eq. (2.104) states that
The response of a linear time-variant system to a complex exponential
function of frequency f is the same complex exponential function multiplied
by a constant coefficient H(f)
An alternative definition of the transfer function
∞
H( f ) =
y (t )
x(t ) x ( t ) =exp( j 2πft )
(2.105) x(t ) = ∫
−∞
X ( f ) exp( j 2πft )df (2.106)
∞
x(t ) = ∆f →0
lim ∑ X ( f ) exp( j 2πft )∆f
f = k∆f k = −∞
(2.107)
∞
y (t ) = ∆f →0 ∑ H ( f ) X ( f ) exp( j 2πft )∆f
lim
f = k∆f k = −∞
∞
= ∫−∞
H ( f ) X ( f ) exp( j 2πft )df (2.108)
81
82. The Fourier transform of the output signal y(t)
Y ( f ) = H ( f ) X ( f ) (2.109)
The Fourier transform of the output is equal to the product of the frequency
response of the system and the Fourier transform of the input
The response y(t) of a linear time-invariant system of impulse response h(t) to an
arbitrary input x(t) is obtained by convolving x(t) with h(t), in accordance with Eq.
(2.93)
The convolution of time functions is transformed into the multiplication of their
Fourier transforms
H ( f ) = H ( f ) exp[ jβ ( f )] (2.110)
Amplitude response or magnitude response Phase or phase response
H ( f ) = H (− f ) β ( f ) = − β (− f )
82
83. In some applications it is preferable to work with the logarithm of H(f)
ln H ( f ) = α ( f ) + jβ ( f ) (2.111)
α ( f ) = ln H ( f ) (2.112)
α ' ( f ) = 20 log10 H ( f ) (2.113)
The gain in decible [dB]
α ' ( f ) = 8.69α ( f ) (2.114)
83
85. Paley-Wiener Criterion
The frequency-domain equivalent of the causality requirement
α( f )
∞
∫
−∞ 1 + f
2
df < ∞ (2.115)
85
86. 2.7 Ideal Low-Pass Filters
Filter
A frequency-selective system that is used to limit the spectrum of a signal to
some specified band of frequencies
The frequency response of an ideal low-pass filter condition
The amplitude response of the filter is a constant inside the passband -B≤f
≤B
The phase response varies linearly with frequency inside the pass band of the
filter
exp(− j 2πft0 ), − B ≤ f ≤ B
H( f ) = (2.116)
0, f >B
86
87. Evaluating the inverse Fourier
∫
B
transform of the transfer function h(t ) = exp[ j 2πf (t − t0 )]df (2.117)
−B
of Eq. (2.116)
sin[ j 2πB (t − t0 )]
h(t ) =
π (t − t0 )
= 2 B sin c[2B(t - t 0 )] (2.118)
We are able to build a causal filter
that approximates an ideal low-pass
filter,
with the approximation improving
with increasing delay t0
sin c[2B(t - t 0 )] << 1, for t < 0
87
88. Pulse Response of Ideal Low-Pass Filters
The impulse response of Eq.(2.118) and the response of the filter
h(t ) = 2 B sin c(2 Bt ) (2.119)
∞
y (t ) = ∫
−∞
x(τ )h(t − τ )dτ
∫
T /2
= 2B sin c[2 B (t − τ )]dτ
−T / 2
sin[ 2πB (t − τ )]
∫
T /2
= 2B dτ (2.120)
−T / 2
2πB (t − τ )
λ = 2πB(t − τ )
1 sin λ
2 πB ( t +T / 2 )
y (t ) = ∫
π 2πB ( t −T / 2 ) λ
dλ
1 2πB ( t +T / 2 ) sin λ 2 πB ( t −T / 2 ) sin λ
=
π0 ∫
λ
dλ − 0 ∫
dλ
λ
1
= {Si[2πB (t + T / 2)] − Si[2πB (t − T / 2)]} (2.121)
π
88
89. Sine integral Si(u)
sin x
∫
u
Si(u ) = dx (2.122)
0 x
An oscillatory function of u, having odd symmetry about the origin u=0.
It has its maxima and minima at multiples of π.
It approaches the limiting value (π/2) for large positive values of u.
89
90. The maximum value of Si(u) occurs at umax= π and is equal to
π
1.8519 = (1.179) × ( )
2
The filter response y(t) has maxima and minima at
T 1
t max =± ±
2 2B
1
y (t max ) = [Si(π ) − Si(π − 2πBT )]
π
1
= [Si(π ) + Si(2πBT − π )]
π
π
Odd symmetric property of the sine integral Si(2πBT − π ) = (1 ± ∆ )
2
1
Si(π ) = (1.179)(π / 2) y (t max ) = (1.179 + 1 ± ∆ )
2
1
≈ 1.09 ± ∆ (2.123)
2
90
91. For BT>>1, the fractional deviation ∆ has a very small value
The percentage overshoot in the filter response is approximately 9 percent
The overshoot is practically independent of the filter bandwidth B
91
96. When using an ideal low-pass filter
We must use a time-bandwidth product BT>> 1 to ensure that the waveform of the
filter input is recognizable from the resulting output.
A value of BT greater than unity tends to reduce the rise time as well as decay time
of the filter pulse response.
Approximation of Ideal Low-Pass Filters
The two basic steps involved in the design of filter
1. The approximation of a prescribed frequency response by a realizable transfer
function
2. The realization of the approximating transfer function by a physical device.
the approximating transfer function H’(s) is a rational function
H ' ( s ) = H ( f ) j 2πf = s
( s − z1 )( s − z 2 ) ⋅ ⋅ ⋅ ( s − z m ) Re([ pi ]) < 0, for all i
=K
( s − p1 )( s − p2 ) ⋅ ⋅ ⋅ ( s − pn )
96
97. Minimum-phase systems
A transfer function whose poles and zeros are all restricted to lie inside the
left hand of the s-plane.
Nonminimum-phase systems
Transfer functions are permitted to have zeros on the imaginary axis as well
as the right half of the s-plane.
Basic options to realization
Analog filter
With inductors and capacitors
With capacitors, resistors, and operational amplifiers
Digital filter
These filters are built using digital hardware
Programmable ; offering a high degree of flexibility in design
97
98. 2.8 Correlation and Spectral Density
: Energy Signals
Autocorrelation Function
Autocorrelation function of the energy signal x(t) for a large τ as
∞
Rx (τ ) = ∫
−∞
x(t ) x * (t − τ )dτ (2.124)
The energy of the signal x(t)
The value of the autocorrelation function Rx(τ) for τ=0
∞
R x ( 0) = ∫
2
x(t ) dt
−∞
98
99. Energy Spectral Density
The energy spectral density is a nonnegative real-valued quantity for all f,
even though the signal x(t) may itself be complex valued.
Ψx ( f ) = X ( f )
2
(2.125)
Wiener-Khitchine Relations for Energy Signals
The autocorrelation function and energy spectral density form a Fourier-
transform pair
∞
Ψx ( f ) = ∫
−∞
Rx (τ ) exp(− j 2πfτ )dτ (2.126)
∞
Rx (τ ) = ∫ −∞
Ψx ( f ) exp( j 2πfτ )df (2.127)
99
100. 1. By setting f=0
The total area under the curve of the complex-valued autocorrelation function of a
complex-valued energy signal is equal to the real-valued energy spectral at zero
frequency
∞
∫−∞
Rx (τ )dτ = Ψx (0)
2. By setting τ=0
The total area under the curve of the real-valued energy spectral density of an erergy
signal is equal to the total energy of the signal.
∞
∫−∞
Ψx ( f )d f = Rx (0)
100
102. Effect of Filtering on Energy Spectral Density
When an energy signal is transmitted through a linear time-invariant
filter,
The energy spectral density of the resulting output equals the energy
spectral density of the input multiplied by the squared amplitude response
of the filter.
Y( f ) = H ( f )X ( f )
Ψy ( f ) = H ( f ) Ψx ( f ) (2.129)
2
An indirect method for evaluating the effect of linear time-invariant
filtering on the autocorrelation function of an energy signal
1. Determine the Fourier transforms of x(t) and h(t), obtaining X(f) and H(f),
respectively.
2. Use Eq. (2.129) to determine the energy spectral density Ψy(f) of the output
y(t).
3. Determine Ry(τ) by applying the inverse Fourier transform to Ψy(f) obtained
under point 2.
102
109. Interpretation of the Energy Spectral Density
The filter is a band-pass filter whose amplitude response is
∆f ∆f
1, f c − ≤ f ≤ fc +
H( f ) = 2 2
0, otherwise
(2.134)
The amplitude spectrum of the filter output
Y( f ) = H( f ) X ( f )
∆f ∆f
X ( fc ) , fc − ≤ f ≤ fc +
= 2 2
0,
otherwise (2.135)
The energy spectral density of the filter output
∆f ∆f
Ψx ( f c ), f c − ≤ f ≤ fc +
Ψy ( f ) = 2 2
0,
otherwise (2.136)
Fig. 2.33
109
111. The energy of the filter output is
∞ ∞
E y = ∫ Ψy ( f )df = 2 ∫ Ψy ( f )df
−∞ 0
The energy spectral density of an energy signal for any frequency f
The energy per unit bandwidth, which is contributed by frequency components of the
signal around the frequency f
E y ≈ 2Ψx ( f c )∆f (2.137)
Ey
Ψx ( f c ) ≈ (2.138)
2∆f
Fig. 2.33
111
112. Cross-Correlation of Energy Signals
The cross-correlation function of the pair
∞
Rxy (τ ) = ∫ −∞
x(t ) y * (t − τ )dt (2.139)
The energy signals x(t) and y(t) are said to be orthogonal over the entire time
domain
If Rxy(0) is zero
∞
∫−∞
x(t ) y * (t )dt = 0 (2.140)
The second cross-correlation function
∞
Ryx (τ ) = ∫−∞
y (t ) x * (t − τ )dt (2.141)
Rxy (τ ) = Ryx (−τ ) (2.142)
*
112
113. The respective Fourier transforms of the cross-correlation functions
Rxy(τ) and Ryx(τ)
∞
∫
Ψxy ( f ) =
−∞
Rxy (τ ) exp(− j 2πfτ )dτ (2.143)
∞
Ψyx (f)=∫ Ryx (τ ) exp(− j 2πfτ )dτ (2.144)
−∞
With the correlation theorem
Ψxy ( f ) = X ( f )Y * ( f ) (2.145)
Ψyx ( f ) = Y ( f ) X * ( f ) (2.146)
The properties of the cross-spectral density
1. Unlike the energy spectral density, cross-spectral density is complex valued
in general.
2. Ψxy(f)= Ψ*yx(f) from which it follows that, in general, Ψxy(f)≠ Ψyx(f)
113
114. 2.9 Power Spectral Density
The average power of a signal is
1 T
P = lim ∫ x(t ) dt (2.147)
2
T →∞
2T −T
P<∞
Truncated version of the signal x(t)
t
xT (t ) = x(t )rect
2T
x(t ), − T ≤ t ≤ T
=
0, otherwise (2.148)
xT (t ) ⇔ X T ( f )
114
115. The average power P in terms of xT(t)
1 ∞
P = lim ∫
2
xT (t ) dt (2.149)
T →∞ −∞
2T
∞ ∞
∫ xT (t ) dt = ∫
2 2
X T ( f ) df
−∞ −∞
1 ∞
P = lim ∫
2
X T ( f ) df (2.150)
T →∞ −∞
2T
∞ 1 2
P= ∫ lim
− ∞ T → ∞ 2T
X T ( f ) df
(2.151)
Power spectral density or Power spectrum
1
S x ( f ) = lim
2
XT ( f ) (2.152)
T →∞
2T
The total area under the curve of the power spectral density of a power signal
is equal to the average power of that signal. ∞
P= ∫−∞
S x ( f )df (2.153)
115
118. 2.10 Numerical Computation of the Fourier Transform
The Fast Fourier transform algorithm
Derived from the discrete Fourier transform
Frequency resolution is defined by fs 1 1
∆f = = = (2.160)
N NTs T
Discrete Fourier transform (DFT) and inverse discrete Fourier transform of
the sequence gn as
g n = g (nTs ) (2.161)
j 2π
N −1
Gk = ∑ g n exp − kn , k = 0,1,..., N − 1 (2.162)
n =0 N
1 N −1 j 2π
g n = ∑ Gk exp kn , n = 0,1,..., N − 1 (2.163)
N k =0 N
118
119. Interpretations of the DFT and the IDFT
For the interpretation of the IDFT process,
We may use the scheme shown in Fig. 2.34(b)
j 2π 2π 2π
exp kn = cos kn + j sin kn
N N N
2π 2π
= cos kn , sin kn , k = 0,1,..., N − 1 (2.164)
N N
At each time index n, an output is formed by summing the weighted complex
generator outputs
Fig. 2.34
119
122. Fast Fourier Transform Algorithms
Be computationally efficient because they use a greatly reduced number of
arithmetic operations
Defining the DFT of gn
N −1
Gk = ∑
n =0
g nW nk , k = 0,1,..., N − 1 (2.165)
j 2π
W = exp − (2.166)
N
W N = exp(− j 2π ) = 1
W N / 2 = exp(− jπ ) = −1
W ( k +lN )( n + mN ) = W kn , for m, l = 0,±1,±2,...
N = 2L
122
123. We may divide the data sequence into two parts
( N / 2 ) −1 N −1
Gk = ∑g W
n =0
n
nk
+ ∑g W
n= N / 2
n
nk
( N / 2 ) −1 ( N / 2 ) −1
= ∑n=0
g nW + nk
∑n =0
g n + N / 2W k ( n + N / 2 )
( N / 2 ) −1
= ∑(g
n =0
n + g n + N / 2W kN / 2 )W kn , k = 0,1,..., N − 1 (2.167)
W kN / 2 = (−1) k
For the case of even k, k=2l,
xn = g n + g n + N / 2 (2.168)
j 4π
W = exp −
2
( N / 2 ) −1 N
∑
N
G2 l = xn (W 2 ) ln , l = 0,1,..., − 1 (2.169) j 2π
n =0 2 = exp −
N /2
123
124. For the case of odd k N
k = 2l + 1, l = 0,1,..., −1
2
yn = g n − g n + N / 2 (2.170)
( N / 2 ) −1
G2 l +1 = ∑n =0
ynW ( 2 l +1) n
( N / 2 ) −1
∑
N
= [ ynW n ](W 2 ) ln , l = 0,1,..., − 1 (2.171)
n =0 2
The sequences xn and yn are themselves related to the original data
sequence
Thus the problem of computing an N-point DFT is reduced to that of
computing two (N/2)-point DFTs.
Fig. 2.35
124
127. Important features of the FFT algorithm
1. At each stage of the computation, the new set of N complex numbers
resulting from the computation can be stored in the same memory
locations used to store the previous set.(in-place computation)
2. The samples of the transform sequence Gk are stored in a bit-reversed
order. To illustrate the meaning of this latter terminology, consider
Table 2.2 constructed for the case of N=8.
Fig. 2.36
127
129. Computation of the IDFT
Eq. (2.163) may rewrite in terms of the complex parameter W
N −1
∑G W
1
gn = k
− kn
, n = 0,1,..., N − 1 (2.172)
N k =0
N −1
Ng =
*
n ∑
k =0
Gk*W kn , 0,1,..., N − 1 (2.173)
129
130. 2.11 Theme Example : Twisted Pairs for Telephony
Fig. 2.39
The typical response of a twisted pair with lengths of 2 to 8 kilometers
Twisted pairs run directly form the central office to the home with one pair
dedicated to each telephone line. Consequently, the transmission lines can
be quite long
The results in Fig. assume a continuous cable. In practice, there may be
several splices in the cable, different gauge cables along different parts of
the path, and so on. These discontinuities in the transmission medium will
further affect the frequency response of the cable.
Fig. 2.39
130
131. We see that, for a 2-km cable, the frequency response is quite flat over the
voice band from 300 to 3100 Hz for telephonic communication. However,
for 8-km cable, the frequency response starts to fall just above 1 kHz.
The frequency response falls off at zero frequency because there is a
capacitive connection at the load and the source. This capacitive connection
is put to practical use by enabling dc power to be transported along the
cable to power the remote telephone handset.
It can be improved by adding some reactive loading
Typically 66 milli-henry (mH) approximately every 2 km.
The loading improves the frequency response of the circuit in the range
corresponding to voice signals without requiring additional power.
Disadvantage of reactive loading
Their degraded performance at high frequency
Fig. 2.39
131
133. 2.12 Summary and Discussion
Fourier Transform
A fundamental tool for relating the time-domain and frequency-domain
descriptions of a deterministic signal
Inverse relationship
Time-bandwidth product of a energy signal is a constant
Linear filtering
Convolution of the input signal with the impulse response of the filter
Multiplication of the Fourier transform of the input signal by the
transfer function of the filter
Correlation
Autocorrelation : a measure of similarity between a signal and a
delayed version of itself
Cross-correlation : when the measure of similarity involves a pair of
different signals
133
134. Spectral Density
The Fourier transform of the autocorrelation function
Cross-Spectral Density
The Fourier transform of the cross-correlation function
Discrete Fourier Transform
Standard Fourier transform by uniformly sampling both the input signal
and the output spectrum
Fast Fourier transform Algorithm
A powerful computation tool for spectral analysis and linear filtering
134
