[1.1] - University of Babylon · Stremler, “Introduction to communication system”, Addison-Wesley Publishing Com., 1999. 5. B. P. Lathi, “signal processing and linear systems”,

[1.1]

Dr. Ahmed A. Alrekaby

UCOMMUNICATION SYSTEM ELEMENTS

UReferences

1. B. P. Lathi, “Modern Digital and Analog Communication Systems”, 3rd Ed., 1998. 2. R. Ziemer, W. Tranter, “Principles Of Communications, Systems, Modulation, and Noise” , 5th Ed., 2002. 3. Hwei P. Hsu, “theory and problems of analog and digital communications”, McGRAW-HILL, Schaume’s

outline series.1993. 4. F. G. Stremler, “Introduction to communication system”, Addison-Wesley Publishing Com., 1999. 5. B. P. Lathi, “signal processing and linear systems”, Berkeley Cambridge Press, 1998. 6. J. G. Proakis, “Digital Communications”, 4th Ed., 2001. U1. COMMUNICATION SYSTEM ELEMENTS

Figure 1.1 shows a commonly used model for a single-link communication system. Although it suggests a system for communication between two remotely located points, this block diagram is also applicable to remote sensing systems, such as radar or sonar, in which the system input and output may be located at the same site. Regardless of the particular application and configuration, all information transmission systems invariably involve three major subsystems-a transmitter, the channel, and a receiver.

UInput Transducer:U The wide variety of possible sources of information results in many different forms for messages. Regardless of their exact form, however, messages may be categorized as analog or digital. The former may be modeled as functions of a continuous-time variable (e.g., pressure, temperature, speech, music), whereas the latter consists of discrete symbols (e.g., written text). Almost invariably, the message produced by a source must be converted by a transducer to a form suitable for the particular type of communication system employed. For example, in electrical communications, speech waves are converted by a microphone to voltage variations. Such a converted message is referred to as the message signal.

UTransmitter:U The purpose of the transmitter is to couple the message to the channel. Although it is not uncommon to find the input transducer directly coupled to the transmission medium, as for example in some intercom systems, it is often necessary to modulate a carrier wave with the signal from the input transducer. Modulation is the systematic variation of some attribute of the carrier, such as amplitude, phase, or frequency, in accordance with a function of the message signal. There are several reasons for using a carrier and modulating it. Important ones are

(1) For ease of radiation. (2) To reduce noise and interference. (3) For channel assignment. (4) For multiplexing or transmission of several messages over a single channel. (5) To overcome equipment limitations. UChannel:U The channel can have many different forms; the most familiar, perhaps, is the channel that exists between the transmitting antenna of a commercial radio station and the receiving antenna of a radio. In this channel, the transmitted signal propagate through the atmosphere, or free space, to the receiving antenna. However, it is not uncommon to find the transmitter hard-wired to the receiver, as in most local telephone systems. This channel is vastly different from the radio example. However, all channels have one thing in common: the signal undergoes degradation from transmitter to receiver. Although this degradation may occur at any point of the communication system block diagram, it is customarily associated with the channel alone. This degradation often results from noise and other undesired signals

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or interference but also may include other distortion effects as well, such as fading signal levels, multiple transmission paths, and filtering

UReceiver:U The receiver's function is to extract the desired message from the received signal at the channel output and to convert it to a form suitable for the output transducer. Although amplification may be one of the first operations performed by the receiver, especially in radio communications, where the received signal may be extremely weak, the main function of the receiver is to demodulate the received signal. Often it is desired that the receiver output be a scaled, possibly delayed, version of the message signal at the modulator input, although in some cases a more general function of the input message is desired. However, as a result of the presence of noise and distortion, this operation is less than ideal.

UOutput Transducer:U The output transducer completes the communication system. This device converts the electric signal at its input into the form desired by the system user. Perhaps the most common output transducer is a loudspeaker. However, there are many other examples, such as tape recorders, personal computers, meters, and cathode-ray tubes.

Fig.1.1

U2. CLASSIFICATION OF SIGNALS

A signal is a function representing a physical quantity. Mathematically, a signal is represented as a function of an independent variable t. usually t represents time. Thus a signal is denoted by x(t).

U2.1 Continuous-Time and Discrete-Time Signals

A signal x(t) is a continuous-time signal if t is a continuous variable. If t is a discrete variable, that is, x(t) is defined at discrete times, then x(t) is a discrete-time signal. Since a discrete-time signal is defined at discrete times, it is often identified as a sequence of numbers, denoted by {x(n)} or x[n], where n = integer.

U2.2 Analog and Digital Signals

If a continuous-time signal x(t) can take on any values in the continuous interval (a, b), where a may be −∞ and b may be +∞, then the continuous-time signal x(t) is called an analog signal. If a discrete-time signal x(t) can take only a finite number of distinct values, then we call this signal a digital signal. The discrete-time signal x[n] is often formed by sampling a continuous-time signal x(t) such that x[n]=x(nTs), where Ts is the sampling interval.

Input message

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U2.3 Real and Complex Signals

A signal x(t) is a real signal if its value is a real number and is a complex signal if its value is a complex number.

U2.4 Deterministic and random signals

Deterministic signals are those signals whose values are completely specified for any given time. Random signals are those signals that take random values at any given time, and these must be characterized statistically.

U2.5 Energy and Power Signals

The normalized energy content E of a signal x(t) is defined as

𝐸𝐸 = � |𝑥𝑥(𝑡𝑡)|2𝑑𝑑𝑡𝑡∞

−∞ (2.1)

The normalized average power P of a signal x(t) is defined as

𝑃𝑃 = lim𝑇𝑇→∞

1𝑇𝑇� |𝑥𝑥(𝑡𝑡)|2𝑑𝑑𝑡𝑡∞

−∞ (2.2)

• If 0 < E < ∞, that is, if E is finite (so P = 0), then x(t) is referred to as an energy signal. • If E = ∞, but 0 < P < ∞, that is, P is finite, then x(t) is referred to as a power signal.

UExample 2.1:U determine whether the signal x(t) = 𝑒𝑒−𝑎𝑎|𝑡𝑡|is power or energy signals or neither.

USol.U 𝑥𝑥(𝑡𝑡) = 𝑒𝑒−𝑎𝑎|𝑡𝑡| = �𝑒𝑒−𝑎𝑎𝑡𝑡 𝑡𝑡 > 0𝑒𝑒𝑎𝑎𝑡𝑡 𝑡𝑡 < 0

�

𝐸𝐸 = ∫ [𝑥𝑥(𝑡𝑡)]2𝑑𝑑𝑡𝑡 = ∫ 𝑒𝑒−2𝑎𝑎|𝑡𝑡|𝑑𝑑𝑡𝑡 = 2∫ 𝑒𝑒−2𝑎𝑎𝑡𝑡∞0 𝑑𝑑𝑡𝑡 = 1

𝑎𝑎< ∞∞

−∞∞−∞ , thus, x(t) is an energy signal.

UH.W U: repeat Example 2.1 with x(t)=A[u(t+a)-u(t-a)], a>0 and x(t)=tu(t).

U2.6 Periodic and Non-periodic Signals

A signal x(t) is periodic if there is a positive number To such that

x(t+nTo) = x(t) (2.3)

The smallest positive number To is called the period, and the reciprocal of the period is called the fundamental frequency fo

𝑓𝑓𝑜𝑜 =1𝑇𝑇𝑜𝑜

ℎ𝑒𝑒𝑒𝑒𝑡𝑡𝑒𝑒 (𝐻𝐻𝑒𝑒) (2.4)

x(t)

0 t

𝑒𝑒−𝑎𝑎𝑡𝑡 𝑒𝑒𝑎𝑎𝑡𝑡

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u(t)

Any signal for which there is no value of To satisfying Eq.(2.3) is said to be nonperiodic or aperiodic, A periodic signal is a power signal if its energy content per period is finite, and then the average power of the signal need only be calculated over a period.

UExample 2.2:U Let x1(t) and x2(t) be periodic signals with periods T1 and T2, respectively: Under what conditions is the sum

x(t) =x1(t) +x2(t) periodic, and what is the period of x(t) .if it is periodic?

USol.U From Eq. (2.3)

x1(t) = x1(t + mT1) m an integer x2(t) = x2(t +nT2) n an integer

If, therefore,T1, and T2 are such that mT1 = nT2 = T

then, x(t + T) =x1(t + T) +x2(t + T) =x1(t) +x2(t) =x(t) that is, x(t) is periodic. Thus, the condition for x(t) to be periodic is

𝑇𝑇1

𝑇𝑇2=𝑛𝑛𝑚𝑚

= rational number

The smallest common period is the least common multiple of T1 and T2. If the ratio T1/T2 is an irrational number, then the signals x1(t) and x2(t) do not have a common period and x(t) cannot be periodic.

To check 𝑥𝑥(𝑡𝑡) = 𝑐𝑐𝑜𝑜𝑐𝑐 �13𝑡𝑡� + 𝑐𝑐𝑠𝑠𝑛𝑛(1

4𝑡𝑡) for periodicity, 𝑐𝑐𝑜𝑜𝑐𝑐 �1

3𝑡𝑡� is periodic with period T1=6π,

and 𝑐𝑐𝑠𝑠𝑛𝑛(14𝑡𝑡) is periodic with period T2=8π. Since 𝑇𝑇1

𝑇𝑇2= 6𝜋𝜋

8𝜋𝜋= 3

4 is a rational number, 𝑥𝑥(𝑡𝑡) is

periodic with period T = 4T1 =3T2 = 24π.

UH.W:U Is the following signal periodic? If so, find their period.

𝑥𝑥(𝑡𝑡) = 𝑐𝑐𝑜𝑜𝑐𝑐(𝑡𝑡) + 2𝑐𝑐𝑠𝑠𝑛𝑛(√2𝑡𝑡)

U2.7 Singularity Functions

An important subclass of non periodic signals in communication theory is the singularity functions (or, as they are sometimes called, the generalized functions).

U2.7.1 Unit Step Function

The unit step function u(t) is defined as

𝑢𝑢(𝑡𝑡) = �1 𝑡𝑡 > 00 𝑡𝑡 < 0 (2.5) �

Note that it is discontinuous at t = 0 and that the value at t = 0 is undefined.

0 t Fig.2.1

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U2.7.2 Unit Impulse Function

The unit impulse function, also known as the Dirac delta function, δ(t) is not an ordinary function and is defined in terms of the following process:

� 𝜙𝜙(𝑡𝑡)𝛿𝛿(𝑡𝑡) 𝑑𝑑𝑡𝑡 = 𝜙𝜙(0) (2.6)∞

−∞

where φ(t) is any test function continuous at t = 0.

Some additional properties of δ(t) are

� 𝜙𝜙(𝑡𝑡)𝛿𝛿(𝑡𝑡 − 𝑡𝑡𝑜𝑜) 𝑑𝑑𝑡𝑡 = 𝜙𝜙(𝑡𝑡𝑜𝑜) (2.7)∞

−∞

𝛿𝛿(𝑎𝑎𝑡𝑡) =1

|𝑎𝑎| 𝛿𝛿(𝑡𝑡) (2.8)

𝛿𝛿(−𝑡𝑡) = 𝛿𝛿(𝑡𝑡) (2.9)

𝑥𝑥(𝑡𝑡)𝛿𝛿(𝑡𝑡) = 𝑥𝑥(0)𝛿𝛿(𝑡𝑡) (2.10)

𝑥𝑥(𝑡𝑡)𝛿𝛿(𝑡𝑡 − 𝑡𝑡𝑜𝑜) = 𝑥𝑥(𝑡𝑡𝑜𝑜)𝛿𝛿(𝑡𝑡 − 𝑡𝑡𝑜𝑜) (2.11)

An alternate definition of δ(t) is provided by the following two conditions:

� 𝛿𝛿(𝑡𝑡 − 𝑡𝑡𝑜𝑜) 𝑑𝑑𝑡𝑡 = 1 𝑡𝑡1 < 𝑡𝑡𝑜𝑜 < 𝑡𝑡2

𝑡𝑡2

𝑡𝑡1

(2.12)

𝛿𝛿(𝑡𝑡 − 𝑡𝑡𝑜𝑜) = 0 𝑡𝑡 ≠ 𝑡𝑡𝑜𝑜 (2.13)

Conditions in Eq.(2.12) and (2.13) correspond to the intuitive notion of a unit impulse as the limit of a suitably chosen conventional function having unity area in an infinitesimally small width. For convenience, δ(t) is shown schematically in Fig. 2.2.

If g(t) is a generalized function, its derivative g'(t) is defined by the following relation:

� g(𝑡𝑡)́∞

−∞𝜙𝜙(𝑡𝑡)𝑑𝑑𝑡𝑡 = −� g(𝑡𝑡)

∞

−∞�́�𝜙(𝑡𝑡)𝑑𝑑𝑡𝑡 (2.14)

By using Eq. (2.14), the derivative of u(t) can be shown to be δ(t); that is,

𝛿𝛿(𝑡𝑡) = �́�𝑢(𝑡𝑡) =𝑑𝑑𝑢𝑢(𝑡𝑡)𝑑𝑑𝑡𝑡

(2.15)

UExample 2.3:U Evaluate the integral, ∫ (𝑡𝑡2 + 𝑐𝑐𝑜𝑜𝑐𝑐𝜋𝜋𝑡𝑡)𝛿𝛿(𝑡𝑡 − 1) 𝑑𝑑𝑡𝑡∞−∞

δ(t)

0 t

Fig.2.2

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USol.U ∫ (𝑡𝑡2 + 𝑐𝑐𝑜𝑜𝑐𝑐𝜋𝜋𝑡𝑡)𝛿𝛿(𝑡𝑡 − 1) 𝑑𝑑𝑡𝑡∞−∞ = �𝑡𝑡2 + 𝑐𝑐𝑜𝑜𝑐𝑐𝜋𝜋𝑡𝑡|𝑡𝑡=1 = 1 + 𝑐𝑐𝑜𝑜𝑐𝑐𝜋𝜋 = 1 − 1 = 0

UH.W:U Evaluate the following integrals;

(a) ∫ 𝑒𝑒−𝑡𝑡𝛿𝛿(2𝑡𝑡 − 2)𝑑𝑑𝑡𝑡∞−∞

(b) ∫ 𝑒𝑒−2𝑡𝑡 �́�𝛿(𝑡𝑡)𝑑𝑑𝑡𝑡∞−∞

U3. COMPLEX EXPONENTIAL FOURIER SERIES

Let x(t) be a periodic signal with period To. Then we define the complex exponential Fourier series of x(t) as

𝑥𝑥(𝑡𝑡) = � 𝑐𝑐𝑛𝑛𝑒𝑒𝑗𝑗𝑛𝑛𝜔𝜔𝑜𝑜𝑡𝑡∞

𝑛𝑛=−∞

(3.1)

where 𝜔𝜔𝑜𝑜 = 2𝜋𝜋 𝑇𝑇𝑜𝑜 = 2𝜋𝜋𝑓𝑓𝑜𝑜 ,⁄ which is called the fundamental angular frequency. The coefficients cn are called the Fourier coefficients, and they are given by

𝑐𝑐𝑛𝑛 =1𝑇𝑇𝑜𝑜� 𝑥𝑥(𝑡𝑡)𝑡𝑡𝑜𝑜+𝑇𝑇𝑜𝑜

𝑡𝑡𝑜𝑜𝑒𝑒−𝑗𝑗𝑛𝑛 𝜔𝜔𝑜𝑜𝑡𝑡𝑑𝑑𝑡𝑡 (3.2)

U3.1 Frequency Spectra

If the periodic signal x(t) is real, then

𝑐𝑐𝑛𝑛 = |𝑐𝑐𝑛𝑛 |𝑒𝑒𝑗𝑗𝜃𝜃𝑛𝑛 𝑐𝑐−𝑛𝑛 = 𝑐𝑐𝑛𝑛∗ = |𝑐𝑐𝑛𝑛 |𝑒𝑒−𝑗𝑗𝜃𝜃𝑛𝑛 (3.3)

The asterisk (*) indicates the complex conjugate. Note that

|𝑐𝑐−𝑛𝑛 | = |𝑐𝑐𝑛𝑛 | 𝜃𝜃−𝑛𝑛 = −𝜃𝜃𝑛𝑛 (3.4)

A plot of |𝑐𝑐𝑛𝑛 | versus the angular frequency ω = 2 π f is called the amplitude spectrum of the periodic signal x(t). A plot of 𝜃𝜃𝑛𝑛 versus ω is called the phase spectrum of x(t). These are referred to as frequency spectra of x(t). Since the index n assumes only integers, the frequency spectra of a periodic signal exist only at the discrete frequencies nωo. These are therefore referred to as discrete frequency Spectra or line spectra. From Eq. (3.4) we see that the amplitude spectrum is an even function of ω and the phase spectrum is an odd function of ω.

UExample 3.1:U Find and sketch the magnitude spectra for the periodic square pulse train signal x(t) shown in Figure beside for (a) d=T/4 and (b) d=T/8.

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The magnitude spectrum for this case is shown in Figure below

(b) d=T/8, nωod/2=nπd/T=nπ/8

The magnitude spectrum for this case is shown in Figure below

UH.W:U If x1(t) and x2(t) are periodic signals with period T and their complex Fourier series expressions are

𝑥𝑥1(𝑡𝑡) = ∑ 𝑑𝑑𝑛𝑛𝑒𝑒𝑗𝑗𝑛𝑛 𝜔𝜔0𝑡𝑡∞

𝑛𝑛=−∞ 𝑥𝑥2(𝑡𝑡) = ∑ 𝑔𝑔𝑛𝑛𝑒𝑒𝑗𝑗𝑛𝑛 𝜔𝜔0𝑡𝑡∞𝑛𝑛=−∞ 𝜔𝜔𝑜𝑜 = 2𝜋𝜋

𝑇𝑇

show that the signal x(t) =x1(t) x2(t) is periodic with the same period T and can be expressed

𝑥𝑥(𝑡𝑡) = ∑ 𝑐𝑐𝑛𝑛𝑒𝑒𝑗𝑗𝑛𝑛 𝜔𝜔0𝑡𝑡∞𝑛𝑛=−∞

where cn is given by 𝑐𝑐𝑛𝑛 = ∑ 𝑑𝑑𝑘𝑘𝑔𝑔𝑛𝑛−𝑘𝑘∞

𝑘𝑘=−∞

U3.2 Power Content of a Periodic Signal

The power content of a periodic signal x(t) with period To is defined as the mean-square value over a period:

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𝑃𝑃 =1𝑇𝑇𝑜𝑜� |𝑥𝑥(𝑡𝑡)|2𝑑𝑑𝑡𝑡𝑇𝑇𝑜𝑜 2⁄

−𝑇𝑇𝑜𝑜 2⁄ (3.5)

U3.3 Parseval’s Theorem for the Fourier Series

Parseval's theorem for the Fourier series states that if x(t) is a periodic signal with period To, then

1𝑇𝑇𝑜𝑜� |𝑥𝑥(𝑡𝑡)|2𝑑𝑑𝑡𝑡𝑇𝑇𝑜𝑜 2⁄

−𝑇𝑇𝑜𝑜 2⁄ = � |𝑐𝑐𝑛𝑛 |2

∞

𝑛𝑛=−∞

(3.6)

UH.W:U Verify Eq.(3.6).

U4. FOURIER TRANSFORM

To generalize the Fourier series representation Eq.(3.1) to a representation valid for non-periodic signals in the frequency domain, we introduce the Fourier transform. Let x(t) be a non-periodic signal. Then the Fourier transform of x(t), symbolized by ℱ, is defined by

𝑋𝑋(𝜔𝜔) = ℱ[𝑥𝑥(𝑡𝑡)] = � 𝑥𝑥(𝑡𝑡)𝑒𝑒−𝑗𝑗𝜔𝜔𝑡𝑡 𝑑𝑑𝑡𝑡 (4.1)∞

−∞

The inverse Fourier transform of X(ω), symbolized by ℱ−1, is defined by

𝑥𝑥(𝑡𝑡) = ℱ−1[𝑋𝑋(𝜔𝜔)] =1

2𝜋𝜋� 𝑋𝑋(𝜔𝜔)𝑒𝑒𝑗𝑗𝜔𝜔𝑡𝑡 𝑑𝑑𝜔𝜔 (4.2)∞

−∞

Equations (4.1) and (4.2) are often called the Fourier transform pair. Writing X(ω) in terms of amplitude and phase as

𝑋𝑋(𝜔𝜔) = |𝑋𝑋(𝜔𝜔)|𝑒𝑒𝑗𝑗𝜃𝜃 (𝜔𝜔) (4.3) we can show, for real x(t), that

𝑋𝑋(−𝜔𝜔) = 𝑋𝑋∗(𝜔𝜔) = |𝑋𝑋(𝜔𝜔)|𝑒𝑒−𝑗𝑗𝜃𝜃 (𝜔𝜔) (4.4) or

|𝑋𝑋(−𝜔𝜔)| = |𝑋𝑋∗(𝜔𝜔)| 𝜃𝜃(−𝜔𝜔) = −𝜃𝜃(𝜔𝜔) (4.5) Thus, just as for the complex Fourier series, the amplitude spectrum of x(t), denoted by |𝑋𝑋(𝜔𝜔)|, is an even function of ω, and the phase spectrum θ(ω) is an odd function of ω. These are referred to as Fourier spectra of x(t). Equation (4.4) is the necessary and sufficient condition for x(t) to be real.

U4.1 Properties of the Fourier Transform

We use the notation x(t) ↔X(ω) to denote the Fourier transform pair.

U4.1.1 Linearity (Superposition)

a1x1(t)+a2x2(t) ↔ a1X1(ω)+a2X2(ω) (4.6)

where a1 and a2 are any constants.

U4.1.2 Time Shifting

𝑥𝑥(𝑡𝑡 − 𝑡𝑡𝑜𝑜) ↔𝑋𝑋(ω)𝑒𝑒−𝑗𝑗ω𝑡𝑡𝑜𝑜 (4.7)

U4.1.3 frequency shifting

𝑥𝑥(𝑡𝑡)𝑒𝑒𝑗𝑗𝜔𝜔𝑜𝑜𝑡𝑡↔𝑋𝑋(𝜔𝜔 −𝜔𝜔𝑜𝑜) (4.8)

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U4.1.4 Scaling

𝑥𝑥(𝑎𝑎𝑡𝑡)↔1

|𝑎𝑎|𝑋𝑋 �𝑤𝑤𝑎𝑎� (4.9)

U4.1.5 Time Reversal

x(− t)↔X(− ω) (4.10)

U4.1.6 Duality

X(t) ↔2πx(−ω) (4.11)

U4.1.7 Differentiation

• Time differentiation

�́�𝑥(𝑡𝑡) =𝑑𝑑𝑑𝑑𝑡𝑡𝑥𝑥(𝑡𝑡)↔𝑗𝑗𝜔𝜔𝑋𝑋(𝜔𝜔) (4.12)

• Frequency differentiation

(−𝑗𝑗𝑡𝑡)𝑥𝑥(𝑡𝑡)↔X́(𝜔𝜔) =𝑑𝑑𝑑𝑑𝜔𝜔

𝑋𝑋(𝜔𝜔) (4.13) UExample 4.1:U Find the Fourier transform of the rectangular pulse signal x(t) defined by

𝑥𝑥(𝑡𝑡) = 𝑝𝑝𝑎𝑎(𝑡𝑡) = �1 |𝑡𝑡| < 𝑎𝑎0 |𝑡𝑡| > 𝑎𝑎

�

USol.U 𝑋𝑋(𝜔𝜔) = ∫ 𝑥𝑥(𝑡𝑡)𝑒𝑒−𝑗𝑗𝜔𝜔𝑡𝑡 𝑑𝑑𝑡𝑡∞−∞ = ∫ 𝑒𝑒−𝑗𝑗𝜔𝜔𝑡𝑡 𝑑𝑑𝑡𝑡 = 2𝑐𝑐𝑠𝑠𝑛𝑛 (𝑎𝑎𝜔𝜔 )

𝜔𝜔= 2𝑎𝑎 𝑐𝑐𝑠𝑠𝑛𝑛 (𝑎𝑎𝜔𝜔 )

𝑎𝑎𝜔𝜔𝑎𝑎−𝑎𝑎

UH.W: UVerify Eqs. (4.6) to (4.13).

U4.2 Fourier Transforms of Some Useful Signals

[1.10]


UExample 4.2:U Find the Fourier transform of 𝑥𝑥(𝑡𝑡) = 𝑐𝑐𝑠𝑠𝑛𝑛 (𝑎𝑎𝑡𝑡 )𝜋𝜋𝑡𝑡

USol.U From Example 4.1 we have ℱ[𝑝𝑝𝑎𝑎(𝑡𝑡)] = 2𝜔𝜔𝑐𝑐𝑠𝑠𝑛𝑛(𝑎𝑎𝜔𝜔) and from the duality property

ℱ �2𝑡𝑡𝑐𝑐𝑠𝑠𝑛𝑛(𝑎𝑎𝑡𝑡)� = 2𝜋𝜋𝑝𝑝𝑎𝑎(−𝜔𝜔)

Thus, ℱ �𝑐𝑐𝑠𝑠𝑛𝑛 (𝑎𝑎𝑡𝑡 )𝜋𝜋𝑡𝑡

� = 12𝜋𝜋ℱ �2

𝑡𝑡𝑐𝑐𝑠𝑠𝑛𝑛(𝑎𝑎𝑡𝑡)� = 𝑝𝑝𝑎𝑎(−𝜔𝜔) = 𝑝𝑝𝑎𝑎(𝜔𝜔)

Where 𝑝𝑝𝑎𝑎(𝜔𝜔) = �1 |𝜔𝜔| < 𝑎𝑎0 |𝜔𝜔| > 𝑎𝑎

�

U5. CONVOLUTION

The convolution of two signals x1(t) and x2(t), denoted by x1(t) ∗ x2(t), is a new signal x(t), defined by

𝑥𝑥(𝑡𝑡) = 𝑥𝑥1(𝑡𝑡) ∗ 𝑥𝑥2(𝑡𝑡) = � 𝑥𝑥1(𝜏𝜏)𝑥𝑥2(𝑡𝑡 − 𝜏𝜏)𝑑𝑑𝜏𝜏∞

−∞ (5.1)

U5.1 Properties of Convolution

(5.2)

The Convolution with δ Functions:

(5.3)

U5.2 Convolution Theorem

• Time convolution: Let x1(t) ↔ X1(ω) and x2(t) ↔ X2(ω)

Then x1(t) ∗ x2(t) ↔ X1(ω) X2(ω) (5.4) • Frequency convolution:

x1(t) x2(t) ↔ 12𝜋𝜋

X1(ω) ∗ X2(ω) (5.5) UExample 5.1:U Verify Eq. (5.4) and (5.5).

USol. ULet X(ω)=X1(ω) X2(ω), then using Eq.(4.2)

𝑥𝑥(𝑡𝑡) = ℱ−1[𝑋𝑋(𝜔𝜔)] =1

2𝜋𝜋� 𝑋𝑋1(𝜔𝜔)𝑋𝑋2(𝜔𝜔)𝑒𝑒𝑗𝑗𝜔𝜔𝑡𝑡 𝑑𝑑𝜔𝜔 ∞

−∞

(Commutative property)

(Associative property)

(Distributive property)

X(ω) 1

ω a −a

[1.11]


𝑥𝑥(𝑡𝑡) =1

2𝜋𝜋� � 𝑥𝑥1(𝜏𝜏) 𝑒𝑒−𝑗𝑗𝜔𝜔𝜏𝜏 𝑋𝑋2(𝜔𝜔)𝑒𝑒𝑗𝑗𝜔𝜔𝑡𝑡 𝑑𝑑𝜏𝜏 𝑑𝑑𝜔𝜔

∞

−∞

∞

−∞

= � 𝑥𝑥1(𝜏𝜏) �1

2𝜋𝜋� 𝑋𝑋2(𝜔𝜔)𝑒𝑒𝑗𝑗𝜔𝜔 (𝑡𝑡−𝜏𝜏) 𝑑𝑑𝜔𝜔∞

−∞� 𝑑𝑑𝜏𝜏

∞

−∞

= � 𝑥𝑥1(𝜏𝜏)𝑥𝑥2(𝑡𝑡 − 𝜏𝜏)𝑑𝑑𝜏𝜏 = 𝑥𝑥1(𝑡𝑡) ∗ 𝑥𝑥2(𝑡𝑡)∞

−∞

UExample 5.1:U Find the convolution of the two signals

𝑥𝑥1(𝑡𝑡) = 𝑒𝑒−𝛼𝛼𝑡𝑡𝑢𝑢(𝑡𝑡) and 𝑥𝑥2(𝑡𝑡) = 𝑒𝑒−𝛽𝛽𝑡𝑡 𝑢𝑢(𝑡𝑡), 𝛼𝛼 > 𝛽𝛽 > 0

USol.U The steps involved in the convolution are illustrated in Fig. (5.1) for α = 4 and β =2. Mathematically, we can form the integrand by direct substitution:

𝑥𝑥(𝑡𝑡) = 𝑥𝑥1(𝑡𝑡) ∗ 𝑥𝑥2(𝑡𝑡) = � 𝑒𝑒−𝛼𝛼𝛼𝛼𝑢𝑢(𝛼𝛼)∞

−∞𝑒𝑒−𝛽𝛽(𝑡𝑡−𝛼𝛼)𝑢𝑢(𝑡𝑡 − 𝛼𝛼)𝑑𝑑𝛼𝛼

But

𝑢𝑢(𝛼𝛼)𝑢𝑢(𝑡𝑡 − 𝛼𝛼) = �0, 𝛼𝛼 < 01, 0 < 𝛼𝛼 < 𝑡𝑡0, 𝛼𝛼 > 𝑡𝑡

�

thus

𝑥𝑥(𝑡𝑡) = �0, 𝑡𝑡 < 0

� 𝑒𝑒−𝛽𝛽𝑡𝑡 𝑒𝑒−(𝛼𝛼−𝛽𝛽)𝛼𝛼𝑑𝑑𝛼𝛼 =1

𝛼𝛼 − 𝛽𝛽�𝑒𝑒−𝛽𝛽𝑡𝑡 − 𝑒𝑒−𝛼𝛼𝑡𝑡 �, 𝑡𝑡 ≥ 0

𝑡𝑡

0

�

This result for x(t) is also shown in Figure below

Fig. (5.1)

UExample 5.1:U Find the convolution of the two signals shown in Fig.(5.2a) and Fig.(5.2b). USol.:U Here, f(t) has a simpler mathematical description than that of g(t), so it is preferable to invert f(t). Hence we shall determine g(t)∗f(t) rather than f(t)∗g(t). Thus

c(t) = g(t) ∗ f(t) =∫ 𝑔𝑔(𝜏𝜏)𝑓𝑓(𝑡𝑡 − 𝜏𝜏)𝑑𝑑𝜏𝜏∞

−∞ Fig.(5.2c) shows g(τ) and f(-τ), whereas Fig.(2.5d) shows g(τ) and f(t−τ), which is f(−τ) shifted by t. because the edges of f(−τ) are at τ= −1 and 1, the edges of f(t−τ) are at −1+t and 1+t. the two functions overlaps over the interval (0,1+t), so that

[1.12]


𝑐𝑐(𝑡𝑡) = � 𝑔𝑔(𝜏𝜏)𝑓𝑓(𝑡𝑡 − 𝜏𝜏)𝑑𝑑𝜏𝜏∞

−∞

= �13𝜏𝜏 𝑑𝑑𝜏𝜏

1+𝑡𝑡

0

= 16

(𝑡𝑡 + 1)2 -1≤ t ≤1 (5.6) This situation, depicted in Fig.(2.5d), is valid only for −1≤ t ≤1. For 1< t <2, the situation is as illustrated in Fig.(5.2e). The two functions overlap only over the range −1+t to 1+t. therefore

𝑐𝑐(𝑡𝑡) = �13𝜏𝜏𝑑𝑑𝜏𝜏

1+𝑡𝑡

−1+𝑡𝑡

= 23𝑡𝑡 1 ≤ t ≤2 (5.7)

At the transition point t =1, both Eq.(5.6) and Eq.(5.7) yield a value of 2/3. i.e., c(1)=2/3. For 2≤ t <4 the situation is as shown in Fig.(5.2f). g(τ) and f(t−τ) overlap over the interval from −1+t to 3, so that

𝑐𝑐(𝑡𝑡) = �13

3

−1+𝑡𝑡𝜏𝜏𝑑𝑑𝜏𝜏

= − 16

(𝑡𝑡2 − 2𝑡𝑡 − 8) (5.8) Again, both Eqs. (5.7) and (5.8) apply the transition point t =2. For t ≥ 4, f(t−τ) does not overlap g(τ) as depicted in Fig.(5.2g). Consequently

c(t) = 0 t ≥ 4 (5.9) for t<−1 there is no overlap between the two functions (see Fig.(5.2h), so that

c(t) = 0 t<−1 (5.10) Figures (5.2i) shows c(t) plotted according to Eqs.(5.6) through (5.10)

[1.13]


Fig.(5.2)

UH.W:U Find the convolution of the two signal f(t) and g(t) depicted in Fig. (5.3). Write a MATLAB program to accomplish this task.

Fig(5.3)

U6. CORRELATION

U6.1 Correlation of Energy Signals

Let x1(t) and x2(t) be real-valued energy signals. Then the cross-correlation function Rl2(τ) of x1(t) and x2(t) is defined by

𝑅𝑅12(𝜏𝜏) = � 𝑥𝑥1(𝑡𝑡)𝑥𝑥2(𝑡𝑡 − 𝜏𝜏)𝑑𝑑𝑡𝑡 (6.1)∞

−∞

The autocorrelation function of x1(t) is defined as

𝑅𝑅11(𝜏𝜏) = � 𝑥𝑥1(𝑡𝑡)𝑥𝑥1(𝑡𝑡 − 𝜏𝜏)𝑑𝑑𝑡𝑡 (6.2)∞

−∞

Properties of correlation functions: 𝑅𝑅12(𝜏𝜏) = 𝑅𝑅21(−𝜏𝜏) (6.3) 𝑅𝑅11(𝜏𝜏) = 𝑅𝑅11(−𝜏𝜏) (6.4)

𝑅𝑅11(0) = � [𝑥𝑥1(𝑡𝑡)]2𝑑𝑑𝑡𝑡 = 𝐸𝐸 (6.2)∞

−∞

where E is the normalized energy content of x1(t). UExample 6.1:U find and sketch the autocorrelation function for 𝑥𝑥1(𝑡𝑡) = 𝑒𝑒−𝑎𝑎𝑡𝑡𝑢𝑢(𝑡𝑡) and a>0. USol.

𝑅𝑅11(𝜏𝜏) = � 𝑥𝑥1(𝑡𝑡)𝑥𝑥1(𝑡𝑡 − 𝜏𝜏)𝑑𝑑𝑡𝑡∞

−∞

[1.14]


= � 𝑒𝑒−𝑎𝑎𝑡𝑡𝑢𝑢(𝑡𝑡)𝑒𝑒−𝑎𝑎(𝑡𝑡−𝜏𝜏)𝑢𝑢(𝑡𝑡 − 𝜏𝜏)𝑑𝑑𝑡𝑡∞

−∞

= 𝑒𝑒𝑎𝑎𝜏𝜏 � 𝑒𝑒−2𝑎𝑎𝑡𝑡𝑢𝑢(𝑡𝑡)𝑢𝑢(𝑡𝑡 − 𝜏𝜏)𝑑𝑑𝑡𝑡∞

−∞

For τ > 0, 𝑢𝑢(𝑡𝑡)𝑢𝑢(𝑡𝑡 − 𝜏𝜏) = �1 𝑡𝑡 > 𝜏𝜏0 𝑡𝑡 < 𝜏𝜏

�

Thus, 𝑅𝑅11(𝜏𝜏) = 𝑒𝑒𝑎𝑎𝜏𝜏 ∫ 𝑒𝑒−2𝑎𝑎𝑡𝑡𝑑𝑑𝑡𝑡∞𝜏𝜏 = 1

2𝑎𝑎𝑒𝑒−𝑎𝑎𝜏𝜏 , since 𝑅𝑅11(𝜏𝜏)is an even function of τ, we conclude

that 𝑅𝑅11(𝜏𝜏) = 12𝑎𝑎𝑒𝑒−𝑎𝑎|𝜏𝜏| a > 0

U6.2 Energy Spectral Density

Let R11(τ) be the autocorrelation function of x1(t). Then

𝑆𝑆11(𝜔𝜔) = ℱ[𝑅𝑅11(𝜏𝜏)] = � 𝑅𝑅11(𝜏𝜏)𝑒𝑒−𝑗𝑗𝜔𝜔𝜏𝜏 𝑑𝑑𝜏𝜏∞

−∞ (6.3)

is called the energy spectral density of x1(t). Now taking the inverse Fourier transform of Eq. (6.3), we have

𝑅𝑅11(𝜏𝜏) = ℱ−1[𝑆𝑆11(ω)] =1

2𝜋𝜋� 𝑆𝑆11(𝜔𝜔)𝑒𝑒𝑗𝑗𝜔𝜔𝜏𝜏 𝑑𝑑𝜔𝜔∞

−∞ (6.4)

If x1(t) is real, then we have 𝑆𝑆11(𝜔𝜔) = ℱ[𝑅𝑅11(𝜏𝜏)] = |𝑋𝑋1(𝜔𝜔)|2 (6.5)

and 𝑅𝑅11(𝜏𝜏) = 12𝜋𝜋 ∫ |𝑋𝑋1(𝜔𝜔)|2𝑒𝑒𝑗𝑗𝜔𝜔𝜏𝜏 𝑑𝑑𝜔𝜔∞

−∞ (6.6) setting τ=0, we have

𝑅𝑅11(0) =1

2𝜋𝜋� |𝑋𝑋1(𝜔𝜔)|2𝑑𝑑𝜔𝜔∞

−∞ (6.7)

Thus, from Eq.(6.2)

𝐸𝐸 = � |𝑥𝑥1(𝑡𝑡)|2𝑑𝑑𝑡𝑡∞

−∞=

12𝜋𝜋

� |𝑋𝑋1(𝜔𝜔)|2𝑑𝑑𝜔𝜔∞

−∞ (6.8)

This is the reason why S11(ω) = |𝑋𝑋1(𝜔𝜔)|2 is called the energy spectral density of x1(t). Equation (6.8) is also known as Parseval's theorem for the Fourier transform.

UExample 6.2:U Find the energy of signal f(t)=e-atu(t). Determine the frequency W (rad/s) so that the energy contributed by the spectral components of all frequencies below W is 95% of the signal energy Ef. USol.: Uwe have

𝐸𝐸𝑓𝑓 = � 𝑓𝑓2(𝑡𝑡) 𝑑𝑑𝑡𝑡 = � 𝑒𝑒−2𝑎𝑎𝑡𝑡𝑑𝑑𝑡𝑡 =1

2𝑎𝑎

∞

0

∞

−∞

We can verify this result by Parseval’s theorem. For this signal

𝐹𝐹(𝜔𝜔) =1

𝑗𝑗𝜔𝜔 + 𝑎𝑎

and 𝐸𝐸𝑓𝑓 = 1𝜋𝜋 ∫ |𝐹𝐹(𝜔𝜔)|2𝑑𝑑𝜔𝜔 = 1

𝜋𝜋 ∫1

𝜔𝜔2+𝑎𝑎2 𝑑𝑑𝜔𝜔 = 1𝜋𝜋𝑎𝑎𝑡𝑡𝑎𝑎𝑛𝑛−1 ��𝜔𝜔

𝑎𝑎��

0

∞= 1

2𝑎𝑎∞

0∞

0

the band ω=0 to ω=W contains 95% of the signal energy, that is, 0.95/2a. Therefore,

𝑅𝑅11(𝜏𝜏)

0 τ

[1.15]


0.952𝑎𝑎

=1𝜋𝜋�

𝑑𝑑𝜔𝜔𝜔𝜔2 + 𝑎𝑎2 =

1𝜋𝜋𝑎𝑎

𝑡𝑡𝑎𝑎𝑛𝑛−1 �𝜔𝜔𝑎𝑎�

0

𝑊𝑊=

1𝜋𝜋𝑎𝑎

𝑊𝑊

0𝑡𝑡𝑎𝑎𝑛𝑛−1 𝑊𝑊

𝑎𝑎

or 0.95𝜋𝜋2

= 𝑡𝑡𝑎𝑎𝑛𝑛−1 𝑊𝑊𝑎𝑎⟹ 𝑊𝑊 = 12.706𝑎𝑎 𝑒𝑒𝑎𝑎𝑑𝑑/𝑐𝑐

this result indicates that the spectral components of f(t) in the band from 0 (DC) to 12.706a rad/s (2.02a Hz) contribute 95% of the total signal energy, the remaining spectral components (from 12.706a rad/s to ∞) contribute only 5% of the signal energy.

U6.3 Correlation of Power Signals:

The time-average autocorrelation function 𝑅𝑅�11(𝜏𝜏) of a real-valued power signal x1(t) is defined as

𝑅𝑅�11(𝜏𝜏) = lim𝑇𝑇→∞

1𝑇𝑇� 𝑥𝑥1(𝑡𝑡)𝑥𝑥1(𝑡𝑡 − 𝜏𝜏) 𝑑𝑑𝑡𝑡 (6.9)𝑇𝑇 2⁄

−𝑇𝑇 2⁄

Note that

𝑅𝑅�11(0) = lim𝑇𝑇→∞

1𝑇𝑇� [𝑥𝑥1(𝑡𝑡)]2 𝑑𝑑𝑡𝑡 = 𝑃𝑃1 (6.10)𝑇𝑇 2⁄

−𝑇𝑇 2⁄

If x1(t) is periodic with period To, then

𝑅𝑅�11(𝜏𝜏) =1𝑇𝑇𝑜𝑜� 𝑥𝑥1(𝑡𝑡)𝑥𝑥1(𝑡𝑡 − 𝜏𝜏) 𝑑𝑑𝑡𝑡 (6.11)𝑇𝑇0 2⁄

−𝑇𝑇0 2⁄

U6.4 Power Spectral Density

The power spectral density of x1(t), denoted 𝑆𝑆1̅1(ω), is defined as

𝑆𝑆1̅1(𝜔𝜔) = ℱ[𝑅𝑅�11(𝜏𝜏)] = � 𝑅𝑅�11(𝜏𝜏)∞

−∞𝑒𝑒−𝑗𝑗𝜔𝜔𝜏𝜏 𝑑𝑑𝜏𝜏 (6.12)

Then

𝑅𝑅�11(𝜏𝜏) = ℱ−1[𝑆𝑆1̅1(𝜔𝜔)] =1

2𝜋𝜋� 𝑆𝑆1̅1(𝜔𝜔)∞

−∞𝑒𝑒𝑗𝑗𝜔𝜔𝜏𝜏 𝑑𝑑𝜔𝜔 (6.13)

Setting τ=0, we get

𝑅𝑅�11(0) =1

2𝜋𝜋� 𝑆𝑆1̅1(𝜔𝜔)∞

−∞𝑑𝑑𝜔𝜔 (6.14)

Thus, from Eq. (6.10)

𝑃𝑃1 = lim𝑇𝑇→∞

1𝑇𝑇� [𝑥𝑥1(𝑡𝑡)]2 𝑑𝑑𝑡𝑡 =

12𝜋𝜋

� 𝑆𝑆1̅1(𝜔𝜔)∞

−∞𝑑𝑑𝜔𝜔 (6.15)

𝑇𝑇 2⁄

−𝑇𝑇 2⁄

This is the reason why 𝑆𝑆1̅1(𝜔𝜔) is called the power spectral density of x1(t).

UExample 6.2: UVerify Eq.(6.15) for the sine wave signal x1(t)=A sin(ω1t+φ), where ω1=2π/T1.

USol.U From Eq.(6.11),

𝑅𝑅�11(𝜏𝜏) =1𝑇𝑇1� 𝑥𝑥1(𝑡𝑡)𝑥𝑥1(𝑡𝑡 − 𝜏𝜏) 𝑑𝑑𝑡𝑡 𝑇𝑇1 2⁄

−𝑇𝑇1 2⁄

=𝐴𝐴2

𝑇𝑇1� 𝑐𝑐𝑠𝑠𝑛𝑛(𝜔𝜔1𝑡𝑡 + 𝜙𝜙) 𝑐𝑐𝑠𝑠𝑛𝑛[𝜔𝜔1(𝑡𝑡 − 𝜏𝜏) + 𝜙𝜙]𝑑𝑑𝑡𝑡 𝑇𝑇1 2⁄

−𝑇𝑇1 2⁄

=𝐴𝐴2

2𝑇𝑇1� [𝑐𝑐𝑜𝑜𝑐𝑐(𝜔𝜔1τ) − 𝑐𝑐𝑜𝑜𝑐𝑐(2𝜔𝜔1𝑡𝑡 + 2𝜙𝜙 − 𝜔𝜔1τ)]𝑑𝑑𝑡𝑡 𝑇𝑇1 2⁄

−𝑇𝑇1 2⁄

[1.16]


=𝐴𝐴2

2𝑇𝑇1𝑐𝑐𝑜𝑜𝑐𝑐(𝜔𝜔1τ)� 𝑑𝑑𝑡𝑡 =

𝑇𝑇1 2⁄

−𝑇𝑇1 2⁄

𝐴𝐴2

2𝑐𝑐𝑜𝑜𝑐𝑐(𝜔𝜔1τ)

and, 𝑅𝑅�11(0) = 𝐴𝐴2

2

From Eq.(6.12)

𝑆𝑆1̅1(𝜔𝜔) = ℱ[𝑅𝑅�11(𝜏𝜏)] = ℱ �𝐴𝐴2

2𝑐𝑐𝑜𝑜𝑐𝑐(𝜔𝜔1τ)� =

𝜋𝜋𝐴𝐴2

2𝛿𝛿(𝜔𝜔 − 𝜔𝜔1) +

𝜋𝜋𝐴𝐴2

2𝛿𝛿(𝜔𝜔 + 𝜔𝜔1)

Then 1

2𝜋𝜋� 𝑆𝑆1̅1(𝜏𝜏)∞

−∞𝑑𝑑𝜔𝜔 =

12𝜋𝜋

� �𝜋𝜋𝐴𝐴2

2𝛿𝛿(𝜔𝜔 − 𝜔𝜔1) +

𝜋𝜋𝐴𝐴2

2𝛿𝛿(𝜔𝜔 + 𝜔𝜔1)� 𝑑𝑑𝜔𝜔

∞

−∞

=𝐴𝐴2

4� [𝛿𝛿(𝜔𝜔 − 𝜔𝜔1) + 𝛿𝛿(𝜔𝜔 + 𝜔𝜔1)]𝑑𝑑𝜔𝜔∞

−∞

=𝐴𝐴2

4(1 + 1) =

𝐴𝐴2

2= 𝑃𝑃1

U7. SYSTEM REPRESENTATION AND CLASSIFICATION

U7.1 System Representation:

A system is a mathematical model of a physical process that relates the input signal (source or. excitation signal) to the output signal (response signal). Let x(t) and y(t) be the input and output signals, respectively, of a system. Then the system is viewed as a mapping of x(t) into y(t). Symbolically, this is expressed as

𝑦𝑦(𝑡𝑡) = 𝒯𝒯[𝑥𝑥(𝑡𝑡)] (7.1) where 𝒯𝒯 is the operator that produces output y(t) from input x(t), as illustrated in Fig.7.1.

U7.2 System Classification

U7.2.1 Continuous-Time and Discrete-Time Systems

If the input and output signals x(t) and y(t) are continuous-time signals, then the system is called a continuous-time system. If the input and output signals are discrete-time signals or sequences, then the system is called a discrete-time system.

U7.2.2 Linear Systems

If the operator 𝒯𝒯 in Eq. (7.1) satisfies the following two conditions, then 𝒯𝒯 is called a linear operator and the system represented by 𝒯𝒯 is called a linear system;

• Additivity 𝒯𝒯[𝑥𝑥1(𝑡𝑡) + 𝑥𝑥2(𝑡𝑡)] = 𝒯𝒯[𝑥𝑥1(𝑡𝑡)] + 𝒯𝒯[𝑥𝑥2(𝑡𝑡)] = 𝑦𝑦1(𝑡𝑡) + 𝑦𝑦2(𝑡𝑡) (7.2)

For all input signals x1(t) and x2(t). • Homogeneity

𝒯𝒯[𝑎𝑎𝑥𝑥(𝑡𝑡)] = 𝑎𝑎𝒯𝒯[𝑥𝑥(𝑡𝑡)] = 𝑎𝑎𝑦𝑦(𝑡𝑡) (7.3) For all input signals x(t) and scalar a.

Any system that does not satisfy Eq. (7.2) and/or Eq. (7.3) is classified as a nonlinear system

𝒯𝒯 System x(t) y(t)

Fig.7.1

[1.17]


UExample 7.1U: For each of the following systems, determine whether the system is linear.

Thus, the system represented by (b) is not linear. The system also does not satisfy the homogeneity condition

U7.2.3 Time Invariant Systems

If the system satisfies the following condition, then the system is called a time-invariant or fixed system:

𝒯𝒯[𝑥𝑥(𝑡𝑡 − 𝑡𝑡𝑜𝑜)] = 𝑦𝑦(𝑡𝑡 − 𝑡𝑡0) (7.4)

where to is any real constant. Equation (7.4) indicates that the delayed input gives delayed output. A system which does not satisfy Eq. (7.4) is called a time-varying system.

U7.2.4 Linear Time-Invariant (LTI) Systems:

If the system is linear and time-invariant, then the system is called a linear time-invariant (LTI) system.

U8. IMPULSE RESPONSE AND FREQUENCY RESPONSE

U8.1 Impulse Response

The impulse response h(t) of an LTI system is defined to be the response of the system when the input is δ(t), that is

ℎ(𝑡𝑡) = 𝒯𝒯[𝛿𝛿(𝑡𝑡)] (8.1) The function h(t) is arbitrary, and it need not be zero for t < 0. If

h(t) = 0 for t < 0 (8.2) then the system is called causal.

U8.2 Response to an Arbitrary Input:

The response y(t) of an LTI system to an arbitrary input x(t) can be expressed as the convolution of x(t) and the impulse response h(t) of the system, that is,

to

to to

[1.18]


𝑦𝑦(𝑡𝑡) = 𝑥𝑥(𝑡𝑡) ∗ ℎ(𝑡𝑡) = � 𝑥𝑥(𝜏𝜏)ℎ(𝑡𝑡 − 𝜏𝜏)𝑑𝑑𝜏𝜏 (8.3)∞

−∞

Since the convolution is commutative, we also can express the output as

𝑦𝑦(𝑡𝑡) = ℎ(𝑡𝑡) ∗ 𝑥𝑥(𝑡𝑡) = � ℎ(𝜏𝜏)𝑥𝑥(𝑡𝑡 − 𝜏𝜏)𝑑𝑑𝜏𝜏 (8.4)∞

−∞

U8.3 Response of Causal Systems

From Eqs. (8.2) and (8.3) or (8.4), the response y(t) of a causal LTI system is given by

𝑦𝑦(𝑡𝑡) = � 𝑥𝑥(𝜏𝜏)ℎ(𝑡𝑡 − 𝜏𝜏)𝑑𝑑𝜏𝜏 = � 𝑥𝑥(𝑡𝑡 − 𝜏𝜏)ℎ(𝜏𝜏)𝑑𝑑𝜏𝜏∞

0 (8.5)

𝑡𝑡

−∞

A signal x(t) is called causal if it has zero values for t < 0. Thus if the input x(t) is also causal, then

𝑦𝑦(𝑡𝑡) = � 𝑥𝑥(𝜏𝜏)ℎ(𝑡𝑡 − 𝜏𝜏)𝑑𝑑𝜏𝜏 = � 𝑥𝑥(𝑡𝑡 − 𝜏𝜏)ℎ(𝜏𝜏)𝑑𝑑𝜏𝜏𝑡𝑡

0

𝑡𝑡

0 (8.6)

U8.4 Frequency Response

Applying the time convolution theorem of the Fourier transform, we obtain Y(ω) =X(ω)H(ω) (8.7)

H(ω) is referred to as the frequency response (or transfer function) of the system. Thus

𝐻𝐻(𝜔𝜔) = ℱ[ℎ(𝑡𝑡)] =𝑌𝑌(𝜔𝜔)𝑋𝑋(𝜔𝜔)

(8.8)

U8.5 Distortionless Transmission U For distortionless transmission through a system, we require that the exact input signal shape be reproduced at the output. Therefore, if x(t) is the input signal, the required output is 𝑦𝑦(𝑡𝑡) = 𝐾𝐾𝑥𝑥(𝑡𝑡 − 𝑡𝑡𝑑𝑑) (8.9) where td is the time delay and K is a gain constant. Taking the Fourier transform of both sides of Eq. (8.9), we get 𝑌𝑌(𝜔𝜔) = 𝐾𝐾𝑒𝑒−𝑗𝑗𝜔𝜔 𝑡𝑡𝑑𝑑𝑋𝑋(𝜔𝜔) (8.10) For distortionless transmission the system must have

(8.11) That is, the amplitude of 𝐻𝐻(𝜔𝜔) must be constant over the entire frequency range, and the phase of 𝐻𝐻(𝜔𝜔)must be linear with frequency.

LTI system

1

δ(t)

x(t)

X(ω)

H(ω)

h(t)

y(t)=x(t)∗h(t)

Y(ω)=X(ω)H(ω)

[1.19]


U9. RELATIONSHIP BETWEEN INPUT AND OUTPUT SPECTRAL DENSITIES

Consider an LTI system with frequency response H(ω), input x(t), and output y(t). If x(t) and y(t) are energy signals, then by Eq.(6.5) their energy spectral densities are 𝑆𝑆𝑥𝑥𝑥𝑥 (𝜔𝜔) =|𝑋𝑋(𝜔𝜔)|2and 𝑆𝑆𝑦𝑦𝑦𝑦 (𝜔𝜔) = |𝑌𝑌(𝜔𝜔)|2, respectively. Since Y(ω) = H(ω)X(ω), it follows that

𝑆𝑆𝑦𝑦𝑦𝑦 (𝜔𝜔) = |𝐻𝐻(𝜔𝜔)|2𝑆𝑆𝑥𝑥𝑥𝑥 (𝜔𝜔) (9.1)

A similar relationship holds for power signals and power spectral densities; that is,

𝑆𝑆�̅�𝑦𝑦𝑦 (𝜔𝜔) = |𝐻𝐻(𝜔𝜔)|2𝑆𝑆�̅�𝑥𝑥𝑥 (𝜔𝜔) (9.2)

UExample 9.1:U Consider a system with 𝐻𝐻(ω) = 1 (1 + 𝑗𝑗𝜔𝜔)⁄ and input x(t)=e-2tu(t). Find the energy spectral density of the output.

USol.

𝑥𝑥(𝑡𝑡) = 𝑒𝑒−2𝑡𝑡𝑢𝑢(𝑡𝑡) ↔ 𝑋𝑋(𝜔𝜔) =1

𝑗𝑗𝜔𝜔 + 2

𝑆𝑆𝑥𝑥𝑥𝑥 (𝜔𝜔) = |𝑋𝑋(𝜔𝜔)|2 =1

𝜔𝜔2 + 4

The energy spectral density of the output is

𝑆𝑆𝑦𝑦𝑦𝑦 (𝜔𝜔) = |𝐻𝐻(𝜔𝜔)|2|𝑋𝑋(𝜔𝜔)|2 =1

(𝜔𝜔2 + 1)(𝜔𝜔2 + 4)

Documents

[1.1] - University of Babylon · Stremler, “Introduction to communication system”, Addison-Wesley Publishing Com., 1999. 5. B. P. Lathi, “signal processing and linear systems”,