Fourier Representations for Four Classes of Signals
Transcript of Fourier Representations for Four Classes of Signals
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 Β© 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 chapterβ’ Introductionβ’ Complex Sinusoids and Frequency Response of LTI
Systemsβ’ Fourier Representations for Four Classes of Signalsβ’ Discrete-Time Periodic Signals: The Discrete-Time Fourier
Seriesβ’ Continuous-Time Periodic Signals: The Fourier Seriesβ’ Discrete-Time Nonperiodic Signals: The Discrete-Time
Fourier Transformβ’ Continuous-Time Nonperiodic Signals: The Fourier
Transformβ’ Properties of Fourier Representationsβ’ Linearity and Symmetry Propertiesβ’ Convolution Propertyβ’ Differentiation and Integration Propertiesβ’ Time- and Frequency-Shift Propertiesβ’ Finding Inverse Fourier Transforms by Using Partial-
Fraction Expansionsβ’ Multiplication Propertyβ’ Scaling Propertiesβ’ Parseval Relationshipsβ’ Time-Bandwidth Productβ’ 3.18 Dualityβ’ 3.19 Exploring Concepts with MATLAB 312 2
3.3 Fourier Representations for Four Classes of Signals
β’ Fourier series (FS) applies to continuous-time periodic signals and the discrete-time Fourier series (DTFS) applies to discrete-time periodic signals.
β’ Nonperiodic signals have Fourier transform representations.
β’ Fourier transform (FT) applies to a signal that is continuous in time and nonperiodic.
β’ Discretetime Fourier transform (DTFT) applies to a signal that is discrete in time and nonperiodic.
β’ Table 3.1 illustrates the relationship between the temporal properties of a signal and the appropriate Fourier representation.
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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
β’ A periodic signal represented as a weighted superposition of complex sinusoids
β’ The frequency of each sinusoid must be an integer multiple of the signalβs fundamental frequency.
β’ If x[n] is discrete-time signal with fundamental period N, we may represent x[n] by DTFS
ΰ·π₯[π] = Οππ΄[π]πππΞ©0π (3.4)
β’ where Ξ©0= 2/N is the fundamental frequency of x[n]. The frequency of the k-th sinusoid in the superposition is kΞ©0. If x(t) is continuous-time signal of fundamental period T, represent x(t) by the FS
ΰ·π₯[π] = Οππ΄[π]ππππ€0π‘ (3.5)
β’ 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.
β’ A sinusoid whose frequency is integer multiple of fundamental is a harmonic5
β’ The complex sinusoids πππΞ©0π are
N-periodic in the frequency index k, as
shown by the relationship
ππ(π+π)Ξ©0π = πππΞ©0ππππΞ©0π
= ππ2ππππΞ©0π = πππΞ©0π
Note that ππ2ππ = 1 β (for all) π
β’ We may rewrite (3.4) as
ΰ·π₯[π] = Οπ=0πβ1π΄[π]πππΞ©0π (3.6)
β’ For the case of continuous-time
complex sinusoids, we have
ΰ·π₯[π] = Οπ=βββ π΄[π]ππππ€0π‘ (3.7)
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β’ The series representations are approximations to x[n] and x(t).
β’ For the discrete-time case, the mean-square error (MSE) between the signal and its series representation can be written as:
πππΈ =1
πΟπ=0πβ1 π₯ π β ΰ·π₯ π 2 (3.8)
β’ For the continuous-time case:
πππΈ =1
π0ππ₯(π‘) β ΰ·π₯(π‘) 2 ππ‘ (3.9)
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3.3.2 Nonperiodic Signals: Fourier-Transform Representations
β’ Fourier transform representations are weighted integral of complex sinusoids where the variable of integration is the sinusoidβs frequency.
β’ Discrete-time sinusoids are used to represent discrete-time signals in DTFT, while continuous-time sinusoids used to represent continuous-time signals in FT.
β’ Fourier Transform of continuous-time signals are given by:
ΰ·π₯ π‘ =1
2πΰΆ±
ββ
β
π(ππ€)πππ€π‘ππ€
β’ Here, π(ππ€)/2 represents βweightβ or coefficient applied to a sinusoid of frequency w in FT representation
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β’ The DTFT involves sinusoidal frequencies within a 2 interval, as shown by
ΰ·π₯[π] =1
2πΰΆ±
β
π(ππΞ©)ππΞ©ππ€
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3.4 Discrete-Time Periodic Signals: The Discrete-Time Fourier Series
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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
β’ 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)
β’ are the DTFS coefficients of the signal x[n], where x[n] and X[k] are a DTFS pair and denote this relationship as
β’ 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]
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π₯ π =
π=0
πβ1
π π πππΞ©0π
π π =1
π
π=0
πβ1
π₯ π πβππΞ©0π
π₯ π =π·ππΉπ;π0
π[π]
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Figure 3.5 (p. 203)Time-domain signal for Example 3.2.
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Figure 3.6 (p. 204)Magnitude and phase of the DTFS coefficients for the signal in Fig. 3.5.
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Figure 3.7 (p. 205)Signals x[n] for Problem 3.2.
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Figure 3.8 (p. 206)Magnitude and phase of DTFS coefficients for Example 3.3.
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Figure 3.9 (p. 207)A discrete-time impulse train with period N.
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Figure 3.10 (p. 208)Magnitude and phase of DTFS coefficients for Example 3.5.
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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.
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Figure 3.11 (p. 209)Square wave for Example 3.6.
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Figure 3.13 (p. 211): Signals x[n] for Problem 3.6.29
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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] = Β½ and is not shown. (a) J = 1. (b) J = 3. (c) J = 5. (d) J = 23. (e) J = 25.
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x
x
Fig
ure
3.1
4b
(p
. 21
3)
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Example 3.8: Numerical Analysis of the ECG
β’ Evaluate the DTFS representations of the two electrocardiogram (ECG) waveforms depicted in Figs. 3.15(a) and (b).
β’ Figure 3.15(a) depicts a normal ECG, while Fig. 3.15(b) depicts the ECG of a heart experiencing ventricular tachycardia.
β’ The discrete-time signals are drawn as continuous functions, due to the difficulty of depicting all 2000 values in each case.
β’ Ventricular tachycardia as a serious cardiac rhythm disturbance is characterized by a rapid, regular heart rate of approximately 150 beats per minute.
β’ The signals appear nearly periodic, with only slight variations in the amplitude and length of each period.
β’ The DTFS of one period of each ECG may be computed numerically.
β’ The period of the normal ECG is N = 305, while the period of the ECG showing ventricular tachycardia is N = 421.
β’ One period of each waveform is available. Evaluate the DTFS coefficients of each and plot their magnitude spectrum. 33
Solution 3.8β’ Magnitude spectrum of first 60 DTFS coefficients is depicted in Figs. 3.15(c) and (d).
β’ The normal ECG is dominated by a sharp spike or impulsive feature.
β’ Recall that the DTFS coefficients of an impulse train have constant magnitude, as shown in Example 3.4.
β’ The DTFS coefficients of the normal ECG are approximately constant, exhibiting a gradual decrease in amplitude as the frequency increases.
β’ They also have a small magnitude, since there is relatively little power in the impulsive signal.
β’ 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;
β’ Also, because the ventricular tachycardia ECG has greater power than the normal ECG, the DTFS coefficients have a larger amplitude.
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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.
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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
β’ Continuous-time periodic signals are represented by the Fourier series (FS)
β’ 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
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π₯(π‘) =
π=ββ
β
π π πππw0π‘
π π =1
πΰΆ±0
π
π₯(π‘) πβππw0π‘ππ‘
β’ 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).
β’ FS coefficients are known as a frequency-Domain Representation of x(t)
β’ FS representation is used to analyze the effect of systems on periodic signals.
β’ The infinite series in (3.19) is define as
ΰ·π₯(π‘) =
π=ββ
β
π[π]ππππ€0π‘
β’ Under what conditions does ΰ·π₯(π‘) actually converge to x(t)?
β’ we can state several results. First, if x(t) is square integrableβthat is, if
1
πΰΆ±
0
π‘
π₯(π‘) ππ‘ < β
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β’The MSE between ΰ·π₯(π‘) and x(t) is
1
πΰΆ±
0
π‘
π₯ π‘ β ΰ·π₯ π‘ 2ππ‘ = 0
β’ x(t) has a finite number of maxima and minima in one period.
β’ Pointwise convergence of ΰ·π₯ π‘ to π₯ π‘ is guaranteed at all values of t except those corresponding to discontinuities if the Dirichlet conditions are satisfied:β’ π₯ π‘ is bounded.β’ π₯ π‘ has a finite number of maxima and minima in one
period.β’ π₯ π‘ has a finite number of discontinuities in one period.
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Figure 3.16 (p. 216): Time-domain signal for Example 3.9.
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Figure 3.17 (p. 217)Magnitude and phase spectra for Example 3.9.
β’ 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)
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Figure 3.18 (p. 219)Magnitude and phase spectra for Example 3.11.
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Figure 3.19 (p. 219)Full-wave rectified cosine for Problem 3.8.
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(ππ‘)
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50
Figure 3.20 (p. 220)FS coefficients for Problem 3.9.
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Figure 3.21 (p. 221)Square wave for Example 3.13.
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Figure 3.22a&b (p. 222)The FS coefficients, X[k], β50 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.
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Figure 3.22c (p. 222)
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Figure 3.23 (p. 223)Sinc function sinc(u) = sin(u)/(u)
β’ Function sin(u)/(u) occurs often in Fourier analysis and given the name
π πππ π’ =sin(Οπ’)
Οπ’(3.24)
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β’ Maximum of the function is unity at u = 0, zero crossings occur at integers multiples of u.
β’ Portion of this function between zero crossings at u = Β±1 is known as mainlobe
β’ The smaller ripples outside the mainlobe are termed sidelobes
β’ FS coefficients in are expressed as
π π =2π0π
π πππ π2π0π
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Figure 3.24 (p. 223) Periodic signal for Problem 3.10.
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Trigonometric Fourier Series
β’ 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
π₯ π‘ = π΅ 0 + Οπ=1β π΅ π cos ππ€0π‘ + π΄ π sin(ππ€0π‘) (3.25)
β’ where the coefficients may be obtained from π₯ π‘ , using
π΅[0] =1
π0ππ₯ π‘ ππ‘
π΅[π] =2
π0ππ₯ π‘ cos ππ€0π‘ ππ‘ (3.26)
π΄[π] =2
πΰΆ±0
π
π₯ π‘ sin(ππ€0π‘)ππ‘
β’ where B[0] = X[0] represents the time-averaged value of the signal.
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Trigonometric Fourier Seriesβ’ Using Eulerβs formula to expand the cosine and sine functions in (3.26) for π β 0, we
have
π΅ π = π π + π βπ
and (3.27)
π΄ π = π π π β π βπ
Please solve Problem 3.86.
β’ The trigonometric FS coefficients of the square wave studied in Example 3.13 are obtained by substituting Eq. (3.23) π π =
2sin(π2ππ0/π)
π2πinto Eq. (3.27), yielding
π΅ 0 = 2π0/π
π΅ π =2sin(π2ππ0/π)
ππ, π β 0
and
π΄ π = 0
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(3.28)
Trigonometric Fourier Series
β’ The sine coefficients π΄[π] are zero because π₯(π‘) is an even function. Thus, the square wave may be expressed as a sum of harmonically related cosines:
π₯ π‘ =
π=0
β
π΅ π cos ππ€0π‘
β’ 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.
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Problem 3.86
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β’ 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
β’ approximation in the vicinity of the square-wave discontinuities at t = Β±T0 = Β±1/4 is of particular interest. We note that each partial-sum approximation passes through the average value (1/2) of the discontinuity.
β’ On each side of the discontinuity, the approximation exhibits ripple. As Jincreases, the maximum height of the ripples does not appear to change.
β’ In fact, it can be shown that, for any finite J, the maximum ripple is approximately 9% of the discontinuity.
β’ This ripple near discontinuities in partial-sum FS approximations is termed the Gibbs phenomenon
β’ 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 = ΒΌ. The J = 0 term is 0(t) = Β½ 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
.
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Figure 3.25e (p. 226)(e) J = 99.
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69
70
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Figure 3.26 (p. 228)The FS coefficients Y[k], β25 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).
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Figure 3.27 (p. 229)Switching power supply for DC-to-AC conversion.
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Figure 3.28 (p. 229)Switching power supply output waveforms with fundamental frequency 0 = 2/T = 120. (a) On-off switch. (b) Inverting switch.
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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
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3.6 Discrete-Time Nonperiodic Signals:The Discrete-Time Fourier Transform
β’ The DTFT is used to represent a discrete-time nonperiodic signal as a superposition of complex sinusoids.
β’ DTFT representation of time-domain signal involves an integral over frequency
π₯[π] =1
2πβπ(ππΞ©)ππΞ©ππΞ© (3.31)
β’ where
π ππΞ© = Οπ=βββ π₯ π πβπΞ©π (3.32)
β’ is DTFT of signal x[n]. We say that X(ππΞ©) and x[n] are a DTFT pair and write
π₯[π]π·ππΉπ
π ππΞ©
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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.
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Figure 3.30 (p. 233)Example 3.18. (a) Rectangular pulse in the time domain. (b) DTFT in the frequency domain.
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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
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Figure 3.33 (p. 236)Example 3.21. (a) Unit impulse in the frequency domain. (b) Inverse DTFT in the time domain.
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Figure 3.34 (p. 236): Signal x[n] for Problem 3.12.
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Figure 3.35 (p. 237)Frequency-domain signal for Problem 3.13.
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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.
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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.
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3.7 Continuous-Time Nonperiodic Signals: The Fourier Transform
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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
β’ Fourier representation of the signal involves a continuum of Frequencies ranging from -β to β.
β’ FT of continuous-time signal involves an integral over entire frequency interval
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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
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Figure 3.39 (p. 243)Example 3.24. (a) Real time-domain exponential signal. (b) Magnitude spectrum. (c) Phase spectrum.
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Example: Find the Fourier Transform of a complex exponential waveform π π = πππππ
β’ Fourier Transform of a complex exponential
π₯ π‘ = πππ€0π‘ = 1 Γ πππ€0π‘ Φ 2ππΏ π€ β πΉ{πππ€0π‘}
π ππ€ = ΰΆ±ββ
β
πππ€0π‘πβππ€π‘ππ‘ =ΰΆ±ββ
β
πβπ π€βπ€0 π‘ππ‘ = 2ππΏ(π€ β π€0)
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π₯ π‘ = πππ€0π‘
π‘0
1
πΏ(π€ β π€0)
π€0π€
π
π‘0
Example: Find the Fourier Transform of a cosine waveform π π = ππ¨π¬(πππ)
β’ Since πππ€0π‘ Φ 2ππΏ π€ β π€0
β’ Fourier Transform of π₯ π‘ = cos π€0π‘ =1
2πππ€0π‘ +
1
2πβππ€0π‘ β
π ππ€ = ΰΆ±ββ
β 1
2πππ€0π‘πβππ€π‘ππ‘ + ΰΆ±
ββ
β 1
2πβππ€0π‘πβππ€π‘ππ‘
=1
22ππΏ π€ β π€0 +
1
22ππΏ π€ + π€0
= ππΏ π€ β π€0 + ππΏ(π€ + π€0)
Hence
cos π€0π‘πΉπ;π€0
ππΏ π€ β π€0 + ππΏ π€ + π€0
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π€π€0βπ€0
π(ππ€)
0
πππΏ π€ + π€0 ππΏ π€ β π€0
Example: Find the Fourier Transform of a cosine waveform π π = ππ¨π¬(πππ)
β’ Fourier Transform of a complex exponential
π₯ π‘ = πππ€0π‘ = 1 Γ πππ€0π‘ Φ 2ππΏ π€ β πΉ{πππ€0π‘}
π ππ€ = ΰΆ±ββ
β
πππ€0π‘πβππ€π‘ππ‘ =ΰΆ±ββ
β
πβπ π€βπ€0 π‘ππ‘ = 2ππΏ(π€ β π€0)
β’ Fourier Transform of π₯ π‘ = cos π€0π‘ =1
2πππ€0π‘ +
1
2πβππ€0π‘ β
π ππ€ = ΰΆ±ββ
β 1
2πππ€0π‘πβππ€π‘ππ‘ + ΰΆ±
ββ
β 1
2πβππ€0π‘πβππ€π‘ππ‘
=1
22ππΏ π€ β π€0 +
1
22ππΏ π€ + π€0
= ππΏ π€ β π€0 + ππΏ(π€ + π€0)
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π€π€0βπ€0
π(ππ€)
0
π
ππΏ π€ + π€0 ππΏ π€ β π€0
Figure 4.2a (p. 343): FT of a cosine.108
π€
π(ππ€)
π€0βπ€00
π
cos(π€0π‘)
π‘2π
π€0β2π
π€0
cos π€0π‘πΉπ;π€0
ππΏ π€ β π€0 + ππΏ π€ + π€0
cos π€0π‘πΉπ;π€0
= α1/2, π = Β±10, π β Β±1
Figure 4.2b (p. 343): FT of a sine 109
cos(π€0π‘)
π‘
2π
π€0β2π
π€0
sin π€0π‘πΉπ;π€0
πππΏ π€ β π€0 β πππΏ π€ + π€0
sin π€0π‘πΉπ;π€0
= α+π/2 π = +1βπ/2 π = β10, π β Β±1
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π€π€0
βπ€0
π(ππ€)
0
π
βπππΏ π€ + π€0
πππΏ π€ β π€0
π
2π
π€0
2π
π€0
2π
π€0
0
βπ
Figure 4.2.a A real signal in frequency domain transforms into a complex signal in time domain and vice versa
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πΏ(π€ β π€0)
π€0 π€
ππππ€0π‘
π‘
πππ€0π‘ = cos π€0π‘ + πsin(π€ππ‘)
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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).
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Figure 3.42 (p. 246)Example 3.26. (a) Rectangular spectrum in the frequency domain. (b) Inverse FT in the time domain.
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Figure 3.43 (p. 248)Frequency-domain signals for Problem 3.15. (a) Part (d). (b) Part (e).
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π(ππ€)
arg π ππ€ π ππ€π
2
βπ
2
π€
π€ π€
Example: Find the Fourier Transform of a step function π(π).
π₯ π‘ = α0 π‘ < 01 π‘ β₯ 0
The Fourier Transform formula can be written as
π ππ€ = ΰΆ±ββ
β
π₯ π‘ πβππ€π‘ππ‘ =ΰΆ±0
β
π₯ π‘ πβππ€π‘ππ‘ =ΰΆ±0
β
cos π€π‘ ππ‘ β ΰΆ±0
β
sin(π€π‘)ππ‘
Is not defined. However, we can interpret π₯(π‘) as πΌ β 0 of a one-sided exponential
ππΌ π‘ = απβππ‘ π‘ β₯ 0
0 π‘ < 0
for values of π < 0. The Fourier Transform will then be
πΊπΌ ππ€ =1
π + ππ€=
π β ππ€
π2 +π€2=
π
π2 +π€2β
ππ€
π2 +π€2
As π β βπ
π2+π€2 β ππΏ(π€) and βππ€
π2+π€2 β1
ππ€
The Fourier Transform of the unit step function will then be
π ππ€ = ΰΆ±0
β
πβππ€π‘ππ‘ =ππΏ π€ +1
ππ€ 121
122
Example: Find the Fourier Transform of a step function π’(π‘).
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
π(ππ) =sin πππ0
ππ
πππ΅ ππ =
= 10 log10sin πππ0
ππdB
10 log10 1 = 0 ππ΅
10log1010β3=β30ππ΅
Main lobe
Main lobes
131
β’ Which, using frequency f in Hz, may be expressed as,
ππβ² ππ =
23sin πππ0
ππ+ 0.5
23π ππ π π β
1π0
π0
π(π β 1/π0)+ 0.5
23π ππ π π +
1π0
π0
π(π + 1/π0)
β’ The first term in this expression corresponds to the spectrum of the rectangular pulse.
β’ The second and third terms have the exact same shape but are shifted in frequency by Β±1/To.
β’ ,Each of these three terms is depicted on the same graph in Fig. 3.47 for π0 = 1.
β’ 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.
β’ Thus, sum of these three terms has lower sidelobes than that of the spectrum of the rectangular pulse.
β’ The normalized magnitude spectrum in dB, which is shown in Fig. 3.48 is
20log ππβ² ππ /π0
β’ The normalization by π0 removed the dependence of the magnitude on π0.
β’ 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/π0 or π0 = 2 Γ 10β4s.
β’ The corresponding data transmission rate is 5000 bits per second.
β’ 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
ππβ² ππ =
23sin πππ0
ππ+ 0.5
23sin π π β
1π0
π0
π(π β 1/π0)+ 0.5
23sin π π +
1π0
π0
π(π + 1/π0)
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 Β© 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.