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

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 chapter• Introduction• Complex Sinusoids and Frequency Response of LTI

Systems• Fourier Representations for Four Classes of Signals• Discrete-Time Periodic Signals: The Discrete-Time Fourier

Series• Continuous-Time Periodic Signals: The Fourier Series• Discrete-Time Nonperiodic Signals: The Discrete-Time

Fourier Transform• Continuous-Time Nonperiodic Signals: The Fourier

Transform• Properties of Fourier Representations• Linearity and Symmetry Properties• Convolution Property• Differentiation and Integration Properties• Time- and Frequency-Shift Properties• Finding Inverse Fourier Transforms by Using Partial-

Fraction Expansions• Multiplication Property• Scaling Properties• Parseval Relationships• Time-Bandwidth Product• 3.18 Duality• 3.19 Exploring Concepts with MATLAB 312 2

3.3 Fourier Representations for Four Classes of Signals

• Fourier series (FS) applies to continuous-time periodic signals and the discrete-time Fourier series (DTFS) applies to discrete-time periodic signals.

• Nonperiodic signals have Fourier transform representations.

• Fourier transform (FT) applies to a signal that is continuous in time and nonperiodic.

• Discretetime Fourier transform (DTFT) applies to a signal that is discrete in time and nonperiodic.

• Table 3.1 illustrates the relationship between the temporal properties of a signal and the appropriate Fourier representation.

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

Fourier Transform

Discrete[n]

Discrete-Time Fourier Series

(DTFS)

Discrete-Time Fourier Transform

(DTFT)4

3.3.1 Periodic Signals: Fourier Series Representations

• A periodic signal represented as a weighted superposition of complex sinusoids

• The frequency of each sinusoid must be an integer multiple of the signal’s fundamental frequency.

• If x[n] is discrete-time signal with fundamental period N, we may represent x[n] by DTFS

ො𝑥[𝑛] = σ𝑘𝐴[𝑘]𝑒𝑗𝑘Ω0𝑛 (3.4)

• where Ω0= 2/N is the fundamental frequency of x[n]. The frequency of the k-th sinusoid in the superposition is kΩ0. If x(t) is continuous-time signal of fundamental period T, represent x(t) by the FS

ො𝑥[𝑛] = σ𝑘𝐴[𝑘]𝑒𝑗𝑘𝑤0𝑡 (3.5)

• 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.

• A sinusoid whose frequency is integer multiple of fundamental is a harmonic5

• The complex sinusoids 𝑒𝑗𝑘Ω0𝑛 are

N-periodic in the frequency index k, as

shown by the relationship

𝑒𝑗(𝑁+𝑘)Ω0𝑛 = 𝑒𝑗𝑁Ω0𝑛𝑒𝑗𝑘Ω0𝑛

= 𝑒𝑗2𝑛𝑒𝑗𝑘Ω0𝑛 = 𝑒𝑗𝑘Ω0𝑛

Note that 𝑒𝑗2𝜋𝑛 = 1 ∀ (for all) 𝑛

• We may rewrite (3.4) as

ො𝑥[𝑛] = σ𝑘=0𝑁−1𝐴[𝑘]𝑒𝑗𝑘Ω0𝑛 (3.6)

• For the case of continuous-time

complex sinusoids, we have

ො𝑥[𝑛] = σ𝑘=−∞∞ 𝐴[𝑘]𝑒𝑗𝑘𝑤0𝑡 (3.7)

• The series representations are approximations to x[n] and x(t).

• For the discrete-time case, the mean-square error (MSE) between the signal and its series representation can be written as:

𝑀𝑆𝐸 =1

𝑁σ𝑛=0𝑁−1 𝑥 𝑛 − ො𝑥 𝑛 2 (3.8)

• For the continuous-time case:

𝑀𝑆𝐸 =1

𝑇0𝑇𝑥(𝑡) − ො𝑥(𝑡) 2 𝑑𝑡 (3.9)

3.3.2 Nonperiodic Signals: Fourier-Transform Representations

• Fourier transform representations are weighted integral of complex sinusoids where the variable of integration is the sinusoid’s frequency.

• Discrete-time sinusoids are used to represent discrete-time signals in DTFT, while continuous-time sinusoids used to represent continuous-time signals in FT.

• Fourier Transform of continuous-time signals are given by:

ො𝑥 𝑡 =1

2𝜋න

−∞

𝑋(𝑗𝑤)𝑒𝑗𝑤𝑡𝑑𝑤

• Here, 𝑋(𝑗𝑤)/2 represents “weight” or coefficient applied to a sinusoid of frequency w in FT representation

• The DTFT involves sinusoidal frequencies within a 2 interval, as shown by

ො𝑥[𝑛] =1

2𝜋න

𝑋(𝑒𝑗Ω)𝑒𝑗Ω𝑑𝑤

3.4 Discrete-Time Periodic Signals: The Discrete-Time Fourier Series

Time PropertyPeriodic

(t, n)Nonperiodic

(t, n)

Continuous(t)

Fourier Series

Fourier Transform

Discrete[n]

(DTFS)

(DTFT)

3.4 Discrete-Time Periodic Signals: The Discrete-Time Fourier Series

• 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)

• are the DTFS coefficients of the signal x[n], where x[n] and X[k] are a DTFS pair and denote this relationship as

• 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]

𝑥 𝑛 =

𝑘=0

𝑁−1

𝑋 𝑘 𝑒𝑗𝑘Ω0𝑛

𝑋 𝑘 =1

𝑛=0

𝑁−1

𝑥 𝑛 𝑒−𝑗𝑘Ω0𝑛

𝑥 𝑛 =𝐷𝑇𝐹𝑆;𝜔0

𝑋[𝑘]

Figure 3.5 (p. 203)Time-domain signal for Example 3.2.

Figure 3.6 (p. 204)Magnitude and phase of the DTFS coefficients for the signal in Fig. 3.5.

Figure 3.7 (p. 205)Signals x[n] for Problem 3.2.

Figure 3.8 (p. 206)Magnitude and phase of DTFS coefficients for Example 3.3.

Figure 3.9 (p. 207)A discrete-time impulse train with period N.

Figure 3.10 (p. 208)Magnitude and phase of DTFS coefficients for Example 3.5.

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.

Figure 3.11 (p. 209)Square wave for Example 3.6.

Figure 3.13 (p. 211): Signals x[n] for Problem 3.6.29

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] = ½ and is not shown. (a) J = 1. (b) J = 3. (c) J = 5. (d) J = 23. (e) J = 25.

Example 3.8: Numerical Analysis of the ECG

• Evaluate the DTFS representations of the two electrocardiogram (ECG) waveforms depicted in Figs. 3.15(a) and (b).

• Figure 3.15(a) depicts a normal ECG, while Fig. 3.15(b) depicts the ECG of a heart experiencing ventricular tachycardia.

• The discrete-time signals are drawn as continuous functions, due to the difficulty of depicting all 2000 values in each case.

• Ventricular tachycardia as a serious cardiac rhythm disturbance is characterized by a rapid, regular heart rate of approximately 150 beats per minute.

• The signals appear nearly periodic, with only slight variations in the amplitude and length of each period.

• The DTFS of one period of each ECG may be computed numerically.

• The period of the normal ECG is N = 305, while the period of the ECG showing ventricular tachycardia is N = 421.

• One period of each waveform is available. Evaluate the DTFS coefficients of each and plot their magnitude spectrum. 33

Solution 3.8• Magnitude spectrum of first 60 DTFS coefficients is depicted in Figs. 3.15(c) and (d).

• The normal ECG is dominated by a sharp spike or impulsive feature.

• Recall that the DTFS coefficients of an impulse train have constant magnitude, as shown in Example 3.4.

• The DTFS coefficients of the normal ECG are approximately constant, exhibiting a gradual decrease in amplitude as the frequency increases.

• They also have a small magnitude, since there is relatively little power in the impulsive signal.

• 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;

• Also, because the ventricular tachycardia ECG has greater power than the normal ECG, the DTFS coefficients have a larger amplitude.

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.

(t, n)Nonperiodic

(t, n)

Continuous(t)

Fourier Series

Fourier Transform

Discrete[n]

(DTFS)

(DTFT)

3.5 Continuous-Time Periodic Fourier Series

• Continuous-time periodic signals are represented by the Fourier series (FS)

• 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

𝑥(𝑡) =

𝑘=−∞

𝑋 𝑘 𝑒𝑗𝑘w0𝑡

𝑋 𝑘 =1

𝑇න0

𝑥(𝑡) 𝑒−𝑗𝑘w0𝑡𝑑𝑡

• 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).

• FS coefficients are known as a frequency-Domain Representation of x(t)

• FS representation is used to analyze the effect of systems on periodic signals.

• The infinite series in (3.19) is define as

ො𝑥(𝑡) =

𝑘=−∞

𝑋[𝑘]𝑒𝑗𝑘𝑤0𝑡

• Under what conditions does ො𝑥(𝑡) actually converge to x(t)?

• we can state several results. First, if x(t) is square integrable—that is, if

𝑇න

𝑥(𝑡) 𝑑𝑡 < ∞

•The MSE between ො𝑥(𝑡) and x(t) is

𝑇න

𝑥 𝑡 − ො𝑥 𝑡 2𝑑𝑡 = 0

• x(t) has a finite number of maxima and minima in one period.

• Pointwise convergence of ො𝑥 𝑡 to 𝑥 𝑡 is guaranteed at all values of t except those corresponding to discontinuities if the Dirichlet conditions are satisfied:• 𝑥 𝑡 is bounded.• 𝑥 𝑡 has a finite number of maxima and minima in one

period.• 𝑥 𝑡 has a finite number of discontinuities in one period.

Figure 3.16 (p. 216): Time-domain signal for Example 3.9.

Figure 3.17 (p. 217)Magnitude and phase spectra for Example 3.9.

• 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)

Figure 3.18 (p. 219)Magnitude and phase spectra for Example 3.11.

Figure 3.19 (p. 219)Full-wave rectified cosine for Problem 3.8.

(𝜋𝑡)

Figure 3.20 (p. 220)FS coefficients for Problem 3.9.

Figure 3.21 (p. 221)Square wave for Example 3.13.

Figure 3.22a&b (p. 222)The FS coefficients, X[k], –50 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.

Figure 3.22c (p. 222)

Figure 3.23 (p. 223)Sinc function sinc(u) = sin(u)/(u)

• Function sin(u)/(u) occurs often in Fourier analysis and given the name

𝑠𝑖𝑛𝑐 𝑢 =sin(π𝑢)

π𝑢(3.24)

• Maximum of the function is unity at u = 0, zero crossings occur at integers multiples of u.

• Portion of this function between zero crossings at u = ±1 is known as mainlobe

• The smaller ripples outside the mainlobe are termed sidelobes

• FS coefficients in are expressed as

𝑋 𝑘 =2𝑇0𝑇

𝑠𝑖𝑛𝑐 𝑘2𝑇0𝑇

Figure 3.24 (p. 223) Periodic signal for Problem 3.10.

Trigonometric Fourier Series

• 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

𝑥 𝑡 = 𝐵 0 + σ𝑘=1∞ 𝐵 𝑘 cos 𝑘𝑤0𝑡 + 𝐴 𝑘 sin(𝑘𝑤0𝑡) (3.25)

• where the coefficients may be obtained from 𝑥 𝑡 , using

𝐵[0] =1

𝑇0𝑇𝑥 𝑡 𝑑𝑡

𝐵[𝑘] =2

𝑇0𝑇𝑥 𝑡 cos 𝑘𝑤0𝑡 𝑑𝑡 (3.26)

𝐴[𝑘] =2

𝑇න0

𝑥 𝑡 sin(𝑘𝑤0𝑡)𝑑𝑡

• where B[0] = X[0] represents the time-averaged value of the signal.

Trigonometric Fourier Series• Using Euler’s formula to expand the cosine and sine functions in (3.26) for 𝑘 ≠ 0, we

𝐵 𝑘 = 𝑋 𝑘 + 𝑋 −𝑘

and (3.27)

𝐴 𝑘 = 𝑗 𝑋 𝑘 − 𝑋 −𝑘

Please solve Problem 3.86.

• The trigonometric FS coefficients of the square wave studied in Example 3.13 are obtained by substituting Eq. (3.23) 𝑋 𝑘 =

2sin(𝑘2𝜋𝑇0/𝑇)

𝑘2𝜋into Eq. (3.27), yielding

𝐵 0 = 2𝑇0/𝑇

𝐵 𝑘 =2sin(𝑘2𝜋𝑇0/𝑇)

𝑘𝜋, 𝑘 ≠ 0

𝐴 𝑘 = 0

(3.28)

Trigonometric Fourier Series

• The sine coefficients 𝐴[𝑘] are zero because 𝑥(𝑡) is an even function. Thus, the square wave may be expressed as a sum of harmonically related cosines:

𝑥 𝑡 =

𝑘=0

𝐵 𝑘 cos 𝑘𝑤0𝑡

• 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.

Problem 3.86

• 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

• approximation in the vicinity of the square-wave discontinuities at t = ±T0 = ±1/4 is of particular interest. We note that each partial-sum approximation passes through the average value (1/2) of the discontinuity.

• On each side of the discontinuity, the approximation exhibits ripple. As Jincreases, the maximum height of the ripples does not appear to change.

• In fact, it can be shown that, for any finite J, the maximum ripple is approximately 9% of the discontinuity.

• This ripple near discontinuities in partial-sum FS approximations is termed the Gibbs phenomenon

• 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 = ¼. The J = 0 term is 0(t) = ½ and is not shown. (a) J = 1.

Figure 3.25e (p. 226)(e) J = 99.

Figure 3.26 (p. 228)The FS coefficients Y[k], –25 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).

Figure 3.27 (p. 229)Switching power supply for DC-to-AC conversion.

Figure 3.28 (p. 229)Switching power supply output waveforms with fundamental frequency 0 = 2/T = 120. (a) On-off switch. (b) Inverting switch.

(t, n)Nonperiodic

(t, n)

Continuous(t)

Fourier Series

Fourier Transform

Discrete[n]

(DTFS)

(DTFT)

3.6 Discrete-Time Nonperiodic Signals:The Discrete-Time Fourier Transform

• The DTFT is used to represent a discrete-time nonperiodic signal as a superposition of complex sinusoids.

• DTFT representation of time-domain signal involves an integral over frequency

𝑥[𝑛] =1

2𝜋−𝑋(𝑒𝑗Ω)𝑒𝑗Ω𝑛𝑑Ω (3.31)

• where

𝑋 𝑒𝑗Ω = σ𝑛=−∞∞ 𝑥 𝑛 𝑒−𝑗Ω𝑛 (3.32)

• is DTFT of signal x[n]. We say that X(𝑒𝑗Ω) and x[n] are a DTFT pair and write

𝑥[𝑛]𝐷𝑇𝐹𝑇

𝑋 𝑒𝑗Ω

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.

Figure 3.30 (p. 233)Example 3.18. (a) Rectangular pulse in the time domain. (b) DTFT in the frequency domain.

Figure 3.31 (p. 234)Example 3.19. (a) Rectangular pulse in the frequency domain. (b) Inverse DTFT in the time domain.

Figure 3.32 (p. 235)Example 3.20. (a) Unit impulse in the time domain. (b) DTFT of unit impulse in the frequency domain.

Figure 3.33 (p. 236)Example 3.21. (a) Unit impulse in the frequency domain. (b) Inverse DTFT in the time domain.

Figure 3.34 (p. 236): Signal x[n] for Problem 3.12.

Figure 3.35 (p. 237)Frequency-domain signal for Problem 3.13.

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.

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.

3.7 Continuous-Time Nonperiodic Signals: The Fourier Transform

(t, n)Nonperiodic

(t, n)

Continuous(t)

Fourier Series

Fourier Transform

Discrete[n]

(DTFS)

(DTFT)

3.7 Continuous-Time Nonperiodic Signals: The Fourier Transform

• Fourier representation of the signal involves a continuum of Frequencies ranging from -∞ to ∞.

• FT of continuous-time signal involves an integral over entire frequency interval

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

Figure 3.39 (p. 243)Example 3.24. (a) Real time-domain exponential signal. (b) Magnitude spectrum. (c) Phase spectrum.

Example: Find the Fourier Transform of a complex exponential waveform 𝒙 𝒕 = 𝒆𝒋𝒘𝟎𝒕

• Fourier Transform of a complex exponential

𝑥 𝑡 = 𝑒𝑗𝑤0𝑡 = 1 × 𝑒𝑗𝑤0𝑡 ֞ 2𝜋𝛿 𝑤 ∗ 𝐹{𝑒𝑗𝑤0𝑡}

𝑋 𝑗𝑤 = න−∞

𝑒𝑗𝑤0𝑡𝑒−𝑗𝑤𝑡𝑑𝑡 =න−∞

𝑒−𝑗 𝑤−𝑤0 𝑡𝑑𝑡 = 2𝜋𝛿(𝑤 − 𝑤0)

𝑥 𝑡 = 𝑒𝑗𝑤0𝑡

𝛿(𝑤 − 𝑤0)

𝑤0𝑤

Example: Find the Fourier Transform of a cosine waveform 𝒙 𝒕 = 𝐜𝐨𝐬(𝒘𝟎𝒕)

• Since 𝑒𝑗𝑤0𝑡 ֞ 2𝜋𝛿 𝑤 − 𝑤0

• Fourier Transform of 𝑥 𝑡 = cos 𝑤0𝑡 =1

2𝑒𝑗𝑤0𝑡 +

2𝑒−𝑗𝑤0𝑡 ⇒

2𝑒𝑗𝑤0𝑡𝑒−𝑗𝑤𝑡𝑑𝑡 + න

−∞

2𝑒−𝑗𝑤0𝑡𝑒−𝑗𝑤𝑡𝑑𝑡

22𝜋𝛿 𝑤 − 𝑤0 +

22𝜋𝛿 𝑤 + 𝑤0

= 𝜋𝛿 𝑤 − 𝑤0 + 𝜋𝛿(𝑤 + 𝑤0)

cos 𝑤0𝑡𝐹𝑆;𝑤0

𝜋𝛿 𝑤 − 𝑤0 + 𝜋𝛿 𝑤 + 𝑤0

𝑤𝑤0−𝑤0

𝑋(𝑗𝑤)

𝜋𝜋𝛿 𝑤 + 𝑤0 𝜋𝛿 𝑤 − 𝑤0

Example: Find the Fourier Transform of a cosine waveform 𝒙 𝒕 = 𝐜𝐨𝐬(𝒘𝟎𝒕)

• Fourier Transform of a complex exponential

𝑥 𝑡 = 𝑒𝑗𝑤0𝑡 = 1 × 𝑒𝑗𝑤0𝑡 ֞ 2𝜋𝛿 𝑤 ∗ 𝐹{𝑒𝑗𝑤0𝑡}

𝑒𝑗𝑤0𝑡𝑒−𝑗𝑤𝑡𝑑𝑡 =න−∞

𝑒−𝑗 𝑤−𝑤0 𝑡𝑑𝑡 = 2𝜋𝛿(𝑤 − 𝑤0)

• Fourier Transform of 𝑥 𝑡 = cos 𝑤0𝑡 =1

2𝑒𝑗𝑤0𝑡 +

2𝑒−𝑗𝑤0𝑡 ⇒

2𝑒𝑗𝑤0𝑡𝑒−𝑗𝑤𝑡𝑑𝑡 + න

−∞

2𝑒−𝑗𝑤0𝑡𝑒−𝑗𝑤𝑡𝑑𝑡

22𝜋𝛿 𝑤 − 𝑤0 +

22𝜋𝛿 𝑤 + 𝑤0

= 𝜋𝛿 𝑤 − 𝑤0 + 𝜋𝛿(𝑤 + 𝑤0)

𝑤𝑤0−𝑤0

𝑋(𝑗𝑤)

𝜋𝛿 𝑤 + 𝑤0 𝜋𝛿 𝑤 − 𝑤0

Figure 4.2a (p. 343): FT of a cosine.108

𝑋(𝑗𝑤)

𝑤0−𝑤00

cos(𝑤0𝑡)

𝑡2𝜋

𝑤0−2𝜋

𝜋𝛿 𝑤 − 𝑤0 + 𝜋𝛿 𝑤 + 𝑤0

= ቊ1/2, 𝑘 = ±10, 𝑘 ≠ ±1

Figure 4.2b (p. 343): FT of a sine 109

cos(𝑤0𝑡)

𝑤0−2𝜋

sin 𝑤0𝑡𝐹𝑆;𝑤0

𝑗𝜋𝛿 𝑤 − 𝑤0 − 𝑗𝜋𝛿 𝑤 + 𝑤0

sin 𝑤0𝑡𝐹𝑆;𝑤0

= ቐ+𝑗/2 𝑘 = +1−𝑗/2 𝑘 = −10, 𝑘 ≠ ±1

𝑤𝑤0

−𝑤0

𝑋(𝑗𝑤)

−𝑗𝜋𝛿 𝑤 + 𝑤0

𝑗𝜋𝛿 𝑤 − 𝑤0

−𝜋

Figure 4.2.a A real signal in frequency domain transforms into a complex signal in time domain and vice versa

𝛿(𝑤 − 𝑤0)

𝑤0 𝑤

𝜋𝑒𝑗𝑤0𝑡

𝑒𝑗𝑤0𝑡 = cos 𝑤0𝑡 + 𝑗sin(𝑤𝑜𝑡)

Figure 3.40 (p. 244): Example 3.25. (a) Rectangular pulse in the time domain. (b) FT in the frequency domain.

Figure 3.41 (p. 245)Time-domain signals for Problem 3.14. (a) Part (d). (b) Part (e).

Figure 3.42 (p. 246)Example 3.26. (a) Rectangular spectrum in the frequency domain. (b) Inverse FT in the time domain.

Figure 3.43 (p. 248)Frequency-domain signals for Problem 3.15. (a) Part (d). (b) Part (e).

𝑋(𝑗𝑤)

arg 𝑋 𝑗𝑤 𝑋 𝑗𝑤𝜋

−𝜋

𝑤 𝑤

Example: Find the Fourier Transform of a step function 𝒖(𝒕).

𝑥 𝑡 = ቊ0 𝑡 < 01 𝑡 ≥ 0

The Fourier Transform formula can be written as

𝑥 𝑡 𝑒−𝑗𝑤𝑡𝑑𝑡 =න0

cos 𝑤𝑡 𝑑𝑡 − න0

sin(𝑤𝑡)𝑑𝑡

Is not defined. However, we can interpret 𝑥(𝑡) as 𝛼 → 0 of a one-sided exponential

𝑔𝛼 𝑡 = ቊ𝑒−𝑎𝑡 𝑡 ≥ 0

0 𝑡 < 0

for values of 𝑎 < 0. The Fourier Transform will then be

𝐺𝛼 𝑗𝑤 =1

𝑎 + 𝑗𝑤=

𝑎 − 𝑗𝑤

𝑎2 +𝑤2=

𝑎2 +𝑤2−

𝑗𝑤

𝑎2 +𝑤2

As 𝑎 → ∞𝑎

𝑎2+𝑤2 → 𝜋𝛿(𝑤) and −𝑗𝑤

𝑎2+𝑤2 →1

𝑗𝑤

The Fourier Transform of the unit step function will then be

𝑋 𝑗𝑤 = න0

𝑒−𝑗𝑤𝑡𝑑𝑡 =𝜋𝛿 𝑤 +1

𝑗𝑤 121

Example: Find the Fourier Transform of a step function 𝑢(𝑡).

Figure 3.44 (p. 249)Pulse shapes used in BPSK communications. (a) Rectangular pulse. (b) Raised cosine pulse.

Figure 3.45 (p. 249)BPSK signals constructed by using (a) rectangular pulse shapes and (b) raised-cosine pulse shapes.

Figure 3.46 (p. 250)Spectrum of rectangular pulse in dB, normalized by T0.

𝑋(𝑗𝑓) =sin 𝜋𝑓𝑇0

𝜋𝑓

𝑋𝑑𝐵 𝑗𝑓 =

= 10 log10sin 𝜋𝑓𝑇0

𝜋𝑓dB

10 log10 1 = 0 𝑑𝐵

10log1010−3=−30𝑑𝐵

Main lobe

Main lobes

• Which, using frequency f in Hz, may be expressed as,

𝑋𝑐′ 𝑗𝑓 =

23sin 𝜋𝑓𝑇0

𝜋𝑓+ 0.5

23𝑠𝑖𝑛 𝜋 𝑓 −

1𝑇0

𝜋(𝑓 − 1/𝑇0)+ 0.5

23𝑠𝑖𝑛 𝜋 𝑓 +

1𝑇0

𝜋(𝑓 + 1/𝑇0)

• The first term in this expression corresponds to the spectrum of the rectangular pulse.

• The second and third terms have the exact same shape but are shifted in frequency by ±1/To.

• ,Each of these three terms is depicted on the same graph in Fig. 3.47 for 𝑇0 = 1.

• 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.

• Thus, sum of these three terms has lower sidelobes than that of the spectrum of the rectangular pulse.

• The normalized magnitude spectrum in dB, which is shown in Fig. 3.48 is

20log 𝑋𝑐′ 𝑗𝑓 /𝑇0

• The normalization by 𝑇0 removed the dependence of the magnitude on 𝑇0.

• 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/𝑇0 or 𝑇0 = 2 × 10−4s.

• The corresponding data transmission rate is 5000 bits per second.

• 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

𝑋𝑐′ 𝑗𝑓 =

23sin 𝜋𝑓𝑇0

𝜋𝑓+ 0.5

23sin 𝜋 𝑓 −

1𝑇0

𝜋(𝑓 − 1/𝑇0)+ 0.5

23sin 𝜋 𝑓 +

1𝑇0

𝜋(𝑓 + 1/𝑇0)

Figure 3.47.a : Spectrum of the raised-cosine pulse consists of sum of three frequency-shifted sinc functions.134

Figure 3.47.b (p. 252)The spectrum of the raised-cosine pulse consists of a sum of three frequency-shifted sinc functions.

fourier representations for four classes of signals

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