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EENG226Signals and Systems
Prof. Dr. Hasan AMCAElectrical and Electronic Engineering
Department (ee.emu.edu.tr)
Eastern Mediterranean University (emu.edu.tr)
Chapter 3Fourier Representations of Signalsand Linear Time-Invariant Systems
Signals and Systems, 2/E by Simon Haykin and Barry Van VeenCopyright ยฉ 2003 John Wiley & Sons. Inc. All rights reserved. 1
Prepared by Prof. Dr. Hasan AMCA
Fourier Representations for Four Classes of Signals
Chapter 3Fourier Representations of Signals
and Linear Time-Invariant Systems
Objectives of this chapterโข Introductionโข Complex Sinusoids and Frequency Response of LTI
Systemsโข Fourier Representations for Four Classes of Signalsโข Discrete-Time Periodic Signals: The Discrete-Time Fourier
Seriesโข Continuous-Time Periodic Signals: The Fourier Seriesโข Discrete-Time Nonperiodic Signals: The Discrete-Time
Fourier Transformโข Continuous-Time Nonperiodic Signals: The Fourier
Transformโข Properties of Fourier Representationsโข Linearity and Symmetry Propertiesโข Convolution Propertyโข Differentiation and Integration Propertiesโข Time- and Frequency-Shift Propertiesโข Finding Inverse Fourier Transforms by Using Partial-
Fraction Expansionsโข Multiplication Propertyโข Scaling Propertiesโข Parseval Relationshipsโข Time-Bandwidth Productโข 3.18 Dualityโข 3.19 Exploring Concepts with MATLAB 312 2
3.3 Fourier Representations for Four Classes of Signals
โข Fourier series (FS) applies to continuous-time periodic signals and the discrete-time Fourier series (DTFS) applies to discrete-time periodic signals.
โข Nonperiodic signals have Fourier transform representations.
โข Fourier transform (FT) applies to a signal that is continuous in time and nonperiodic.
โข Discretetime Fourier transform (DTFT) applies to a signal that is discrete in time and nonperiodic.
โข Table 3.1 illustrates the relationship between the temporal properties of a signal and the appropriate Fourier representation.
3
Table 3.1 Relationship between time properties of a signal and the appropriate Fourier Representations
Time Property
Periodic (t, n)
Nonperiodic(t, n)
Continuous(t)
Fourier Series
F(S)
Fourier Transform
(FT)
Discrete[n]
Discrete-Time Fourier Series
(DTFS)
Discrete-Time Fourier Transform
(DTFT)4
3.3.1 Periodic Signals: Fourier Series Representations
โข A periodic signal represented as a weighted superposition of complex sinusoids
โข The frequency of each sinusoid must be an integer multiple of the signalโs fundamental frequency.
โข If x[n] is discrete-time signal with fundamental period N, we may represent x[n] by DTFS
เท๐ฅ[๐] = ฯ๐๐ด[๐]๐๐๐ฮฉ0๐ (3.4)
โข where ฮฉ0= 2/N is the fundamental frequency of x[n]. The frequency of the k-th sinusoid in the superposition is kฮฉ0. If x(t) is continuous-time signal of fundamental period T, represent x(t) by the FS
เท๐ฅ[๐] = ฯ๐๐ด[๐]๐๐๐๐ค0๐ก (3.5)
โข Where w0 = 2 / T is the fundamental frequency of x(t). Here, the frequency of the kth sinusoid is kw0 and each sinusoid has a common period T.
โข A sinusoid whose frequency is integer multiple of fundamental is a harmonic5
โข The complex sinusoids ๐๐๐ฮฉ0๐ are
N-periodic in the frequency index k, as
shown by the relationship
๐๐(๐+๐)ฮฉ0๐ = ๐๐๐ฮฉ0๐๐๐๐ฮฉ0๐
= ๐๐2๐๐๐๐ฮฉ0๐ = ๐๐๐ฮฉ0๐
Note that ๐๐2๐๐ = 1 โ (for all) ๐
โข We may rewrite (3.4) as
เท๐ฅ[๐] = ฯ๐=0๐โ1๐ด[๐]๐๐๐ฮฉ0๐ (3.6)
โข For the case of continuous-time
complex sinusoids, we have
เท๐ฅ[๐] = ฯ๐=โโโ ๐ด[๐]๐๐๐๐ค0๐ก (3.7)
6
โข The series representations are approximations to x[n] and x(t).
โข For the discrete-time case, the mean-square error (MSE) between the signal and its series representation can be written as:
๐๐๐ธ =1
๐ฯ๐=0๐โ1 ๐ฅ ๐ โ เท๐ฅ ๐ 2 (3.8)
โข For the continuous-time case:
๐๐๐ธ =1
๐0๐๐ฅ(๐ก) โ เท๐ฅ(๐ก) 2 ๐๐ก (3.9)
7
3.3.2 Nonperiodic Signals: Fourier-Transform Representations
โข Fourier transform representations are weighted integral of complex sinusoids where the variable of integration is the sinusoidโs frequency.
โข Discrete-time sinusoids are used to represent discrete-time signals in DTFT, while continuous-time sinusoids used to represent continuous-time signals in FT.
โข Fourier Transform of continuous-time signals are given by:
เท๐ฅ ๐ก =1
2๐เถฑ
โโ
โ
๐(๐๐ค)๐๐๐ค๐ก๐๐ค
โข Here, ๐(๐๐ค)/2 represents โweightโ or coefficient applied to a sinusoid of frequency w in FT representation
8
โข The DTFT involves sinusoidal frequencies within a 2 interval, as shown by
เท๐ฅ[๐] =1
2๐เถฑ
โ
๐(๐๐ฮฉ)๐๐ฮฉ๐๐ค
9
3.4 Discrete-Time Periodic Signals: The Discrete-Time Fourier Series
10
Table 3.1 Relationship between time properties of a signal and the appropriate Fourier Representations
Time PropertyPeriodic
(t, n)Nonperiodic
(t, n)
Continuous(t)
Fourier Series
F(S)
Fourier Transform
(FT)
Discrete[n]
Discrete-Time Fourier Series
(DTFS)
Discrete-Time Fourier Transform
(DTFT)
3.4 Discrete-Time Periodic Signals: The Discrete-Time Fourier Series
โข The DTFS representation of a periodic signal x[n] with fundamental period N and fundamental frequency 0 = 2N is given by
(3.10)
Where,
(3.11)
โข are the DTFS coefficients of the signal x[n], where x[n] and X[k] are a DTFS pair and denote this relationship as
โข The DTFS coefficients X[k] are termed a frequency-domain representation for x[n], while the coefficients x[n] are the time-domain representation of X[f]
11
๐ฅ ๐ =
๐=0
๐โ1
๐ ๐ ๐๐๐ฮฉ0๐
๐ ๐ =1
๐
๐=0
๐โ1
๐ฅ ๐ ๐โ๐๐ฮฉ0๐
๐ฅ ๐ =๐ท๐๐น๐;๐0
๐[๐]
12
13
14
Figure 3.5 (p. 203)Time-domain signal for Example 3.2.
15
Figure 3.6 (p. 204)Magnitude and phase of the DTFS coefficients for the signal in Fig. 3.5.
16
17
Figure 3.7 (p. 205)Signals x[n] for Problem 3.2.
18
19
20
Figure 3.8 (p. 206)Magnitude and phase of DTFS coefficients for Example 3.3.
21
22
23
Figure 3.9 (p. 207)A discrete-time impulse train with period N.
24
25
Figure 3.10 (p. 208)Magnitude and phase of DTFS coefficients for Example 3.5.
26
Figure 3.12 (p. 211)The DTFS coefficients for the square wave shown in Fig. 3.11, assuming a period N = 50: (a) M = 4. (b) M = 12.
27
Figure 3.11 (p. 209)Square wave for Example 3.6.
28
Figure 3.13 (p. 211): Signals x[n] for Problem 3.6.29
30
Figure 3.14a (p. 213)Individual terms in the DTFS expansion of a square wave (left panel) and the corresponding partial-sum approximations J[n] (right panel). The J = 0 term is
0[n] = ยฝ and is not shown. (a) J = 1. (b) J = 3. (c) J = 5. (d) J = 23. (e) J = 25.
31
x
x
Fig
ure
3.1
4b
(p
. 21
3)
32
Example 3.8: Numerical Analysis of the ECG
โข Evaluate the DTFS representations of the two electrocardiogram (ECG) waveforms depicted in Figs. 3.15(a) and (b).
โข Figure 3.15(a) depicts a normal ECG, while Fig. 3.15(b) depicts the ECG of a heart experiencing ventricular tachycardia.
โข The discrete-time signals are drawn as continuous functions, due to the difficulty of depicting all 2000 values in each case.
โข Ventricular tachycardia as a serious cardiac rhythm disturbance is characterized by a rapid, regular heart rate of approximately 150 beats per minute.
โข The signals appear nearly periodic, with only slight variations in the amplitude and length of each period.
โข The DTFS of one period of each ECG may be computed numerically.
โข The period of the normal ECG is N = 305, while the period of the ECG showing ventricular tachycardia is N = 421.
โข One period of each waveform is available. Evaluate the DTFS coefficients of each and plot their magnitude spectrum. 33
Solution 3.8โข Magnitude spectrum of first 60 DTFS coefficients is depicted in Figs. 3.15(c) and (d).
โข The normal ECG is dominated by a sharp spike or impulsive feature.
โข Recall that the DTFS coefficients of an impulse train have constant magnitude, as shown in Example 3.4.
โข The DTFS coefficients of the normal ECG are approximately constant, exhibiting a gradual decrease in amplitude as the frequency increases.
โข They also have a small magnitude, since there is relatively little power in the impulsive signal.
โข In contrast, the ventricular tachycardia ECG contains smoother features in addition to sharp spikes, and thus the DTFS coefficients have a greater dynamic range, with the low-frequency coefficients containing a large proportion of the total power;
โข Also, because the ventricular tachycardia ECG has greater power than the normal ECG, the DTFS coefficients have a larger amplitude.
34
Figure 3.15 (p. 214)Electrocardiograms for two different heartbeats and the first 60 coefficients of their magnitude spectra. (a) Normal heartbeat. (b) Ventricular tachycardia. (c) Magnitude spectrum for the
normal heartbeat. (d) Magnitude spectrum for
ventricular tachycardia.
35
36
Table 3.1 Relationship between time properties of a signal and the appropriate Fourier Representations
Time PropertyPeriodic
(t, n)Nonperiodic
(t, n)
Continuous(t)
Fourier Series
F(S)
Fourier Transform
(FT)
Discrete[n]
Discrete-Time Fourier Series
(DTFS)
Discrete-Time Fourier Transform
(DTFT)
3.5 Continuous-Time Periodic Fourier Series
โข Continuous-time periodic signals are represented by the Fourier series (FS)
โข We may write the FS of a signal x(t) with fundamental period T and fundamental frequency w0=2/T as
(3.19)
(3.20)
Where:
3.5 Continuous-Time Periodic Fourier Series
37
๐ฅ(๐ก) =
๐=โโ
โ
๐ ๐ ๐๐๐w0๐ก
๐ ๐ =1
๐เถฑ0
๐
๐ฅ(๐ก) ๐โ๐๐w0๐ก๐๐ก
โข From FS coefficients X[k], we may determine x(t) by using (3.19) and from x(t), we may determine X[k] by using (3.20).
โข FS coefficients are known as a frequency-Domain Representation of x(t)
โข FS representation is used to analyze the effect of systems on periodic signals.
โข The infinite series in (3.19) is define as
เท๐ฅ(๐ก) =
๐=โโ
โ
๐[๐]๐๐๐๐ค0๐ก
โข Under what conditions does เท๐ฅ(๐ก) actually converge to x(t)?
โข we can state several results. First, if x(t) is square integrableโthat is, if
1
๐เถฑ
0
๐ก
๐ฅ(๐ก) ๐๐ก < โ
38
โขThe MSE between เท๐ฅ(๐ก) and x(t) is
1
๐เถฑ
0
๐ก
๐ฅ ๐ก โ เท๐ฅ ๐ก 2๐๐ก = 0
โข x(t) has a finite number of maxima and minima in one period.
โข Pointwise convergence of เท๐ฅ ๐ก to ๐ฅ ๐ก is guaranteed at all values of t except those corresponding to discontinuities if the Dirichlet conditions are satisfied:โข ๐ฅ ๐ก is bounded.โข ๐ฅ ๐ก has a finite number of maxima and minima in one
period.โข ๐ฅ ๐ก has a finite number of discontinuities in one period.
39
40
Figure 3.16 (p. 216): Time-domain signal for Example 3.9.
41
Figure 3.17 (p. 217)Magnitude and phase spectra for Example 3.9.
โข As with the DTFS, the magnitude of X[k] is known as the magnitude spectrum of x(t), and phase of X[k] is known as the phase spectrum of x(t)
42
43
44
45
Figure 3.18 (p. 219)Magnitude and phase spectra for Example 3.11.
46
47
Figure 3.19 (p. 219)Full-wave rectified cosine for Problem 3.8.
48
(๐๐ก)
49
50
Figure 3.20 (p. 220)FS coefficients for Problem 3.9.
51
52
Figure 3.21 (p. 221)Square wave for Example 3.13.
53
54
Figure 3.22a&b (p. 222)The FS coefficients, X[k], โ50 k 50, for three square waves. (see Fig. 3.21.) (a) Ts/T = 1/4 . (b) Ts/T = 1/16. (c) Ts/T = 1/64.
55
Figure 3.22c (p. 222)
56
Figure 3.23 (p. 223)Sinc function sinc(u) = sin(u)/(u)
โข Function sin(u)/(u) occurs often in Fourier analysis and given the name
๐ ๐๐๐ ๐ข =sin(ฯ๐ข)
ฯ๐ข(3.24)
57
โข Maximum of the function is unity at u = 0, zero crossings occur at integers multiples of u.
โข Portion of this function between zero crossings at u = ยฑ1 is known as mainlobe
โข The smaller ripples outside the mainlobe are termed sidelobes
โข FS coefficients in are expressed as
๐ ๐ =2๐0๐
๐ ๐๐๐ ๐2๐0๐
58
Figure 3.24 (p. 223) Periodic signal for Problem 3.10.
59
Trigonometric Fourier Series
โข The form of the FS described by Eqs. (3.19) and (3.20) is termed the exponential FS. The trigonometric FS is often useful for real-valued signals and is expressed as
๐ฅ ๐ก = ๐ต 0 + ฯ๐=1โ ๐ต ๐ cos ๐๐ค0๐ก + ๐ด ๐ sin(๐๐ค0๐ก) (3.25)
โข where the coefficients may be obtained from ๐ฅ ๐ก , using
๐ต[0] =1
๐0๐๐ฅ ๐ก ๐๐ก
๐ต[๐] =2
๐0๐๐ฅ ๐ก cos ๐๐ค0๐ก ๐๐ก (3.26)
๐ด[๐] =2
๐เถฑ0
๐
๐ฅ ๐ก sin(๐๐ค0๐ก)๐๐ก
โข where B[0] = X[0] represents the time-averaged value of the signal.
60
Trigonometric Fourier Seriesโข Using Eulerโs formula to expand the cosine and sine functions in (3.26) for ๐ โ 0, we
have
๐ต ๐ = ๐ ๐ + ๐ โ๐
and (3.27)
๐ด ๐ = ๐ ๐ ๐ โ ๐ โ๐
Please solve Problem 3.86.
โข The trigonometric FS coefficients of the square wave studied in Example 3.13 are obtained by substituting Eq. (3.23) ๐ ๐ =
2sin(๐2๐๐0/๐)
๐2๐into Eq. (3.27), yielding
๐ต 0 = 2๐0/๐
๐ต ๐ =2sin(๐2๐๐0/๐)
๐๐, ๐ โ 0
and
๐ด ๐ = 0
61
(3.28)
Trigonometric Fourier Series
โข The sine coefficients ๐ด[๐] are zero because ๐ฅ(๐ก) is an even function. Thus, the square wave may be expressed as a sum of harmonically related cosines:
๐ฅ ๐ก =
๐=0
โ
๐ต ๐ cos ๐๐ค0๐ก
โข This expression offers insight into the manner in which each FS component contributes to the representation of the signal, as is illustrated in the next example.
62
Problem 3.86
63
64
โข so the even-indexed coefficients are zero. The individual terms and partial-sum approximations are depicted in Fig. 3.25. The behavior of the partial-sum
โข approximation in the vicinity of the square-wave discontinuities at t = ยฑT0 = ยฑ1/4 is of particular interest. We note that each partial-sum approximation passes through the average value (1/2) of the discontinuity.
โข On each side of the discontinuity, the approximation exhibits ripple. As Jincreases, the maximum height of the ripples does not appear to change.
โข In fact, it can be shown that, for any finite J, the maximum ripple is approximately 9% of the discontinuity.
โข This ripple near discontinuities in partial-sum FS approximations is termed the Gibbs phenomenon
โข The square wave satisfies the Dirichlet conditions, so we know that the FS approximation ultimately converges to the square wave for all valuesof t, except at the discontinuities. 65
Figure 3.25a (p. 225)Individual terms (top) in the FS expansion of a square wave and the corresponding partial-sum approximations j(t) (bottom). The square wave has period T = 1 and Ts/T = ยผ. The J = 0 term is 0(t) = ยฝ and is not shown. (a) J = 1.
66
x
x
Fig
ure
3.2
5b
-3 (
p. 2
26
)(b
) J
= 3
. (c)
J=
7.
(d)
J=
29
.
67
Figure 3.25e (p. 226)(e) J = 99.
68
69
70
71
Figure 3.26 (p. 228)The FS coefficients Y[k], โ25 k 25, for the RC circuit output in response to a square-wave input. (a) Magnitude spectrum. (b) Phase spectrum. c) One period of the input signal x(t) dashed line) and output signal y(t) (solid line). The output signal y(t) is computed from the partial-sum approximation given in Eq. 3.30).
72
Figure 3.27 (p. 229)Switching power supply for DC-to-AC conversion.
73
74
75
Figure 3.28 (p. 229)Switching power supply output waveforms with fundamental frequency 0 = 2/T = 120. (a) On-off switch. (b) Inverting switch.
76
77
Table 3.1 Relationship between time properties of a signal and the appropriate Fourier Representations
Time PropertyPeriodic
(t, n)Nonperiodic
(t, n)
Continuous(t)
Fourier Series
F(S)
Fourier Transform
(FT)
Discrete[n]
Discrete-Time Fourier Series
(DTFS)
Discrete-Time Fourier Transform
(DTFT)
3.6 Discrete-Time Nonperiodic Signals:The Discrete-Time Fourier Transform
78
3.6 Discrete-Time Nonperiodic Signals:The Discrete-Time Fourier Transform
โข The DTFT is used to represent a discrete-time nonperiodic signal as a superposition of complex sinusoids.
โข DTFT representation of time-domain signal involves an integral over frequency
๐ฅ[๐] =1
2๐โ๐(๐๐ฮฉ)๐๐ฮฉ๐๐ฮฉ (3.31)
โข where
๐ ๐๐ฮฉ = ฯ๐=โโโ ๐ฅ ๐ ๐โ๐ฮฉ๐ (3.32)
โข is DTFT of signal x[n]. We say that X(๐๐ฮฉ) and x[n] are a DTFT pair and write
๐ฅ[๐]๐ท๐๐น๐
๐ ๐๐ฮฉ
79
80
81
Figure 3.29 (p.232)The DTFT of an exponential signal x[n] = ()nu[n]. (a) Magnitude spectrum for = 0.5. (b) Phase spectrum for = 0.5. (c) Magnitude spectrum for = 0.9. (d) Phase spectrum for = 0.9.
82
83
84
Figure 3.30 (p. 233)Example 3.18. (a) Rectangular pulse in the time domain. (b) DTFT in the frequency domain.
85
Figure 3.31 (p. 234)Example 3.19. (a) Rectangular pulse in the frequency domain. (b) Inverse DTFT in the time domain.
86
Figure 3.32 (p. 235)Example 3.20. (a) Unit impulse in the time domain. (b) DTFT of unit impulse in the frequency domain.
87
88
Figure 3.33 (p. 236)Example 3.21. (a) Unit impulse in the frequency domain. (b) Inverse DTFT in the time domain.
89
90
Figure 3.34 (p. 236): Signal x[n] for Problem 3.12.
91
92
93
Figure 3.35 (p. 237)Frequency-domain signal for Problem 3.13.
94
Figure 3.36 (p. 239)The magnitude responses of two simple discrete-time systems. (a) A system that averages successive inputs tends to attenuate high frequencies. (b) A system that forms the difference of successive inputs tends to attenuate low frequencies.
95
96
97
98
Figure 3.37 (p. 241)Magnitude response of the system in Example 3.23 describing multipath propagation. (a) Echo coefficient a = 0.5ej/3. (b) Echo coefficient a = 0.9ej2/3.
99
3.7 Continuous-Time Nonperiodic Signals: The Fourier Transform
100
Table 3.1 Relationship between time properties of a signal and the appropriate Fourier Representations
Time PropertyPeriodic
(t, n)Nonperiodic
(t, n)
Continuous(t)
Fourier Series
F(S)
Fourier Transform
(FT)
Discrete[n]
Discrete-Time Fourier Series
(DTFS)
Discrete-Time Fourier Transform
(DTFT)
3.7 Continuous-Time Nonperiodic Signals: The Fourier Transform
โข Fourier representation of the signal involves a continuum of Frequencies ranging from -โ to โ.
โข FT of continuous-time signal involves an integral over entire frequency interval
101
Figure 3.38 (p. 241)Magnitude response of the inverse system for multipath propagation in Example 3.23. (a) Echo coefficient a = 0.5ej/3. (b) Echo coefficient a = 0.9ej/3
102
103
Figure 3.39 (p. 243)Example 3.24. (a) Real time-domain exponential signal. (b) Magnitude spectrum. (c) Phase spectrum.
104
Example: Find the Fourier Transform of a complex exponential waveform ๐ ๐ = ๐๐๐๐๐
โข Fourier Transform of a complex exponential
๐ฅ ๐ก = ๐๐๐ค0๐ก = 1 ร ๐๐๐ค0๐ก ึ 2๐๐ฟ ๐ค โ ๐น{๐๐๐ค0๐ก}
๐ ๐๐ค = เถฑโโ
โ
๐๐๐ค0๐ก๐โ๐๐ค๐ก๐๐ก =เถฑโโ
โ
๐โ๐ ๐คโ๐ค0 ๐ก๐๐ก = 2๐๐ฟ(๐ค โ ๐ค0)
105
๐ฅ ๐ก = ๐๐๐ค0๐ก
๐ก0
1
๐ฟ(๐ค โ ๐ค0)
๐ค0๐ค
๐
๐ก0
Example: Find the Fourier Transform of a cosine waveform ๐ ๐ = ๐๐จ๐ฌ(๐๐๐)
โข Since ๐๐๐ค0๐ก ึ 2๐๐ฟ ๐ค โ ๐ค0
โข Fourier Transform of ๐ฅ ๐ก = cos ๐ค0๐ก =1
2๐๐๐ค0๐ก +
1
2๐โ๐๐ค0๐ก โ
๐ ๐๐ค = เถฑโโ
โ 1
2๐๐๐ค0๐ก๐โ๐๐ค๐ก๐๐ก + เถฑ
โโ
โ 1
2๐โ๐๐ค0๐ก๐โ๐๐ค๐ก๐๐ก
=1
22๐๐ฟ ๐ค โ ๐ค0 +
1
22๐๐ฟ ๐ค + ๐ค0
= ๐๐ฟ ๐ค โ ๐ค0 + ๐๐ฟ(๐ค + ๐ค0)
Hence
cos ๐ค0๐ก๐น๐;๐ค0
๐๐ฟ ๐ค โ ๐ค0 + ๐๐ฟ ๐ค + ๐ค0
106
๐ค๐ค0โ๐ค0
๐(๐๐ค)
0
๐๐๐ฟ ๐ค + ๐ค0 ๐๐ฟ ๐ค โ ๐ค0
Example: Find the Fourier Transform of a cosine waveform ๐ ๐ = ๐๐จ๐ฌ(๐๐๐)
โข Fourier Transform of a complex exponential
๐ฅ ๐ก = ๐๐๐ค0๐ก = 1 ร ๐๐๐ค0๐ก ึ 2๐๐ฟ ๐ค โ ๐น{๐๐๐ค0๐ก}
๐ ๐๐ค = เถฑโโ
โ
๐๐๐ค0๐ก๐โ๐๐ค๐ก๐๐ก =เถฑโโ
โ
๐โ๐ ๐คโ๐ค0 ๐ก๐๐ก = 2๐๐ฟ(๐ค โ ๐ค0)
โข Fourier Transform of ๐ฅ ๐ก = cos ๐ค0๐ก =1
2๐๐๐ค0๐ก +
1
2๐โ๐๐ค0๐ก โ
๐ ๐๐ค = เถฑโโ
โ 1
2๐๐๐ค0๐ก๐โ๐๐ค๐ก๐๐ก + เถฑ
โโ
โ 1
2๐โ๐๐ค0๐ก๐โ๐๐ค๐ก๐๐ก
=1
22๐๐ฟ ๐ค โ ๐ค0 +
1
22๐๐ฟ ๐ค + ๐ค0
= ๐๐ฟ ๐ค โ ๐ค0 + ๐๐ฟ(๐ค + ๐ค0)
107
๐ค๐ค0โ๐ค0
๐(๐๐ค)
0
๐
๐๐ฟ ๐ค + ๐ค0 ๐๐ฟ ๐ค โ ๐ค0
Figure 4.2a (p. 343): FT of a cosine.108
๐ค
๐(๐๐ค)
๐ค0โ๐ค00
๐
cos(๐ค0๐ก)
๐ก2๐
๐ค0โ2๐
๐ค0
cos ๐ค0๐ก๐น๐;๐ค0
๐๐ฟ ๐ค โ ๐ค0 + ๐๐ฟ ๐ค + ๐ค0
cos ๐ค0๐ก๐น๐;๐ค0
= แ1/2, ๐ = ยฑ10, ๐ โ ยฑ1
Figure 4.2b (p. 343): FT of a sine 109
cos(๐ค0๐ก)
๐ก
2๐
๐ค0โ2๐
๐ค0
sin ๐ค0๐ก๐น๐;๐ค0
๐๐๐ฟ ๐ค โ ๐ค0 โ ๐๐๐ฟ ๐ค + ๐ค0
sin ๐ค0๐ก๐น๐;๐ค0
= แ+๐/2 ๐ = +1โ๐/2 ๐ = โ10, ๐ โ ยฑ1
109
๐ค๐ค0
โ๐ค0
๐(๐๐ค)
0
๐
โ๐๐๐ฟ ๐ค + ๐ค0
๐๐๐ฟ ๐ค โ ๐ค0
๐
2๐
๐ค0
2๐
๐ค0
2๐
๐ค0
0
โ๐
Figure 4.2.a A real signal in frequency domain transforms into a complex signal in time domain and vice versa
110
๐ฟ(๐ค โ ๐ค0)
๐ค0 ๐ค
๐๐๐๐ค0๐ก
๐ก
๐๐๐ค0๐ก = cos ๐ค0๐ก + ๐sin(๐ค๐๐ก)
111
112
Figure 3.40 (p. 244): Example 3.25. (a) Rectangular pulse in the time domain. (b) FT in the frequency domain.
113
Figure 3.41 (p. 245)Time-domain signals for Problem 3.14. (a) Part (d). (b) Part (e).
114
115
116
Figure 3.42 (p. 246)Example 3.26. (a) Rectangular spectrum in the frequency domain. (b) Inverse FT in the time domain.
117
118
119
Figure 3.43 (p. 248)Frequency-domain signals for Problem 3.15. (a) Part (d). (b) Part (e).
120
๐(๐๐ค)
arg ๐ ๐๐ค ๐ ๐๐ค๐
2
โ๐
2
๐ค
๐ค ๐ค
Example: Find the Fourier Transform of a step function ๐(๐).
๐ฅ ๐ก = แ0 ๐ก < 01 ๐ก โฅ 0
The Fourier Transform formula can be written as
๐ ๐๐ค = เถฑโโ
โ
๐ฅ ๐ก ๐โ๐๐ค๐ก๐๐ก =เถฑ0
โ
๐ฅ ๐ก ๐โ๐๐ค๐ก๐๐ก =เถฑ0
โ
cos ๐ค๐ก ๐๐ก โ เถฑ0
โ
sin(๐ค๐ก)๐๐ก
Is not defined. However, we can interpret ๐ฅ(๐ก) as ๐ผ โ 0 of a one-sided exponential
๐๐ผ ๐ก = แ๐โ๐๐ก ๐ก โฅ 0
0 ๐ก < 0
for values of ๐ < 0. The Fourier Transform will then be
๐บ๐ผ ๐๐ค =1
๐ + ๐๐ค=
๐ โ ๐๐ค
๐2 +๐ค2=
๐
๐2 +๐ค2โ
๐๐ค
๐2 +๐ค2
As ๐ โ โ๐
๐2+๐ค2 โ ๐๐ฟ(๐ค) and โ๐๐ค
๐2+๐ค2 โ1
๐๐ค
The Fourier Transform of the unit step function will then be
๐ ๐๐ค = เถฑ0
โ
๐โ๐๐ค๐ก๐๐ก =๐๐ฟ ๐ค +1
๐๐ค 121
122
Example: Find the Fourier Transform of a step function ๐ข(๐ก).
123
124
Figure 3.44 (p. 249)Pulse shapes used in BPSK communications. (a) Rectangular pulse. (b) Raised cosine pulse.
125
126
Figure 3.45 (p. 249)BPSK signals constructed by using (a) rectangular pulse shapes and (b) raised-cosine pulse shapes.
127
128
129
Figure 3.46 (p. 250)Spectrum of rectangular pulse in dB, normalized by T0.
130
๐(๐๐) =sin ๐๐๐0
๐๐
๐๐๐ต ๐๐ =
= 10 log10sin ๐๐๐0
๐๐dB
10 log10 1 = 0 ๐๐ต
10log1010โ3=โ30๐๐ต
Main lobe
Main lobes
131
โข Which, using frequency f in Hz, may be expressed as,
๐๐โฒ ๐๐ =
23sin ๐๐๐0
๐๐+ 0.5
23๐ ๐๐ ๐ ๐ โ
1๐0
๐0
๐(๐ โ 1/๐0)+ 0.5
23๐ ๐๐ ๐ ๐ +
1๐0
๐0
๐(๐ + 1/๐0)
โข The first term in this expression corresponds to the spectrum of the rectangular pulse.
โข The second and third terms have the exact same shape but are shifted in frequency by ยฑ1/To.
โข ,Each of these three terms is depicted on the same graph in Fig. 3.47 for ๐0 = 1.
โข Note that the second and third terms share zero crossings with the first term and have the opposite sign
in the sidelobe region of the first term.
โข Thus, sum of these three terms has lower sidelobes than that of the spectrum of the rectangular pulse.
โข The normalized magnitude spectrum in dB, which is shown in Fig. 3.48 is
20log ๐๐โฒ ๐๐ /๐0
โข The normalization by ๐0 removed the dependence of the magnitude on ๐0.
โข In this case, the peak of the first sidelobe is below -30 dB, so we may satisfy the adjacent channel
interference specifications by choosing the main-lobe to be 20 kHz wide, which implies that
10000 = 2/๐0 or ๐0 = 2 ร 10โ4s.
โข The corresponding data transmission rate is 5000 bits per second.
โข The use of the raised-cosine pulse shape increases the data transmission rate by a factor of five relative
to the rectangular pulse shape in this application. 132
133
๐๐โฒ ๐๐ =
23sin ๐๐๐0
๐๐+ 0.5
23sin ๐ ๐ โ
1๐0
๐0
๐(๐ โ 1/๐0)+ 0.5
23sin ๐ ๐ +
1๐0
๐0
๐(๐ + 1/๐0)
Figure 3.47.a : Spectrum of the raised-cosine pulse consists of sum of three frequency-shifted sinc functions.134
135
Figure 3.47.b (p. 252)The spectrum of the raised-cosine pulse consists of a sum of three frequency-shifted sinc functions.
136
137Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright ยฉ 2003 John Wiley & Sons. Inc. All rights reserved.
Fig
ure
3.4
8 (
p. 2
52
)Sp
ectr
um
of
the
rais
ed-c
osi
ne
pu
lse
in
dB
, no
rmal
ized
by
T0.