2000Communication Fundamentals1 Dr. Charles Surya DE634  6220

2000Communication Fundamentals2 Chapter 1 and 2 Introduction and Signals and Systems

2000Communication Fundamentals3 Communication refers to the conveying of information from one point to another It is a crucial component of information technology, which consists of: generation, transmission, reception, manipulation, storage and display of information Electromagnetic signals are typically used in the transmission of information In the process of transmitting the information, some alterations need to be done on the information-bearing signal to facilitate the transmission process. Upon reception, the inverse operation process needs to be done to retrieve to original signal

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2000Communication Fundamentals5 The encoder chooses the best representation of the information to optimize its detection The decoder performs the reverse operation for the retrieval of the information The modulator produces a varying signal at its output which is proportional in some way to the signal appearing across it input terminals. A sinusoidal modulator may vary the amplitude, frequency or phase of a sinusoidal signal in direct proportion to the voltage input. The encoder and modulator both serve to prepare the signal for more efficient transmission. However, the process of coding is designed to optimize the error-free detection, whereas the process of modulation is designed to impress

2000Communication Fundamentals6 the information signal onto the waveform to be transmitted. The demodulator performs the inverse operation of the modulator to recover the signal in its original form. The transmission medium is the crucial link, which may include the ionosphere, troposphere, free space, or simply a transmission line. Here attenuation, distortion, and noise in the medium are introduced. Noise is any electrical signals that interfere with the error- free reception of the message-bearing signal.

2000Communication Fundamentals7 The 3 basic subsystems of a communication system are indicated by the dashed lines in Fig The transmitter is to prepare the information to be sent in a way that best cope with the restrictions imposed by the channel. The receiver is to perform the inverse of the transmitter operation. The transmitter and the receiver as a pair are specifically designed to combat the deleterious effects of the channel on the information transmission.

2000Communication Fundamentals8 Fig. 1-1 is a simplex system. In many cases it is desirable to maintain 2-way communication. One way to accomplish this is to use the same channel alternately for transmission in each direction. This is called half-duplex.

2000Communication Fundamentals9 The full-duplex, as shown in Fig.1-3, allows simultaneous communication in both directions.

2000Communication Fundamentals10 Signals and Systems A signal is an event capable of starting an action. For our purposes, a signal is defined to be a single-valued function of time and may be real or complex. The complex notation can be used to describe signals in terms of 2 independent variables. Thus, it is convenient for describing 2-D phenomena such as circular motion, plan wave propagation etc. Sinusoids play a major role in the analysis of communication systems e.g.

2000Communication Fundamentals11 Where A is the amplitude,  is the phase and  is the rate of phase change or frequency of the sinusoid in radians/s. The principle of Fourier methods of signal analysis is to break up all signals into summations of sinusoids. This provides a description of a given signal in terms of how the energy and power are distributed in these sinusoidal frequencies.

2000Communication Fundamentals12 Classification of Signals Energy Signals and Power Signals An energy signal is a pulse-like signal that usually exists for only a finite interval of time, or even for an infinite amount of time, at least has a manor portion of its energy concentrated in a finite time interval. The instantaneous power of an electrical signal e(t) is In each case the instantaneous power is proportional to the squared magnitude of the signal. For a 1-Ohm resistance, these equations assume the same form. Thus, it is customary in signal analysis to speak of the instantaneous power associated with a given signal as Watts. The energy dissipated by the signal during a time interval (t 1,t 2 ) is

2000Communication Fundamentals13 We define an energy signal to be one for which

2000Communication Fundamentals14 The average power dissipated by the signal f(t) is A signal with the following property is defined as power signal A periodic signal is one that repeats itself exactly after a fixed period of time otherwise it is an aperiodic signal

2000Communication Fundamentals15 A random signal is one about which there is some degree of uncertainty before it actually occurs, whereas a deterministic signal is one that has no uncertainty in its values.

2000Communication Fundamentals16 Classification of Systems A system is a rule used for assigning an output, g(t), to an input, f(t) i.e. This rule can be in terms of an algebraic operation, a differential and/or integral equation, etc. For two systems connected in cascade the output of the first system forms the input to the second, thus forming a new overall system If a system is linear then superposition applies

2000Communication Fundamentals17 A system is time-invariant if a time shift in the input results in a corresponding time shift in the output so that The output of a time-invariant system depends on time differences and not on absolute values of time. Any system not meeting this requirement is said to be time- varying.

2000Communication Fundamentals18 A physically realizable or causal system cannot have an output response before an arbitrary input function is applied, otherwise it is a physically nonrealizable or noncausal system.

2000Communication Fundamentals19 Orthogonal Functions If we wish to express a function f(t) as a set of numbers, f n, which, when expressed in terms of a properly chosen coordinate space,  n, will specify the function uniquely. It is highly desirable that the set so chosen be a linearly independent set. That is the individual terms are not dependent on each other and that the set is formed by the totality of these terms. Such complete set of orthogonal functions are capable of uniquely representing the function of interest. This is known as the basis function.

2000Communication Fundamentals20 Two complex-valued functions  1 and  2 are orthogonal over the interval (t 1, t 2 ) if Thus if members of a set of complex-valued functions are mutually orthogonal over (t 1, t 2 ) then

2000Communication Fundamentals21 The set of basis functions is said to be “normalized” if If the set is both orthogonal and normalized it is called an orthonormal set. The integral of the product of 2 functions over a given interval is called the inner product of the 2 functions. The square root of the inner product of a function with itself is called the norm.

2000Communication Fundamentals22 f(t) can be approximated by summation of a finite number of term  n The integral-squared error remaining in this approximation after N terms is  n (t) is said to be complete over (t 1, t 2 ) if

2000Communication Fundamentals23 For a complete orthogonal set This relationship is known as the Parseval’s Theorem. Example: A given rectangular function is shown below:

2000Communication Fundamentals24 First we can easily show that sin(n  t) are orthonormal over the interval (0,2) Thus we have where f n is defined as

2000Communication Fundamentals25 Substituting into the equation above we obtain Thus f(t) can be represented by the following series

2000Communication Fundamentals26 The following figure shows the approximation when the function is approximated with 1, 2 and 3 terms.

2000Communication Fundamentals27 The exponential Fourier Series For a set of complex-valued exponential functions where n is an integer and  0 is a constant. The value of n is referred to as the harmonic number or harmonic. Consider the following operation on  n (t)

2000Communication Fundamentals28 Excluding the trivial case where t 2 = t 1, we can force the term within the brackets to zero if we choose in which (n-m) is an integer. Thus, forms an orthogonal set of basis over the interval (t 1, t 2 ) if

2000Communication Fundamentals29 An arbitrary signal f(t) can be expressed as where the coefficients F n are to be determined. It can be shown that the error energy between f(t) and its approximations decreases to zero as the number of terms taken approaches infinity. When a set of  n (t) meets this condition, it is said to be complete It is therefore possible to represent any arbitrary complex-

2000Communication Fundamentals30 valued function with finite energy by a linear combination of complex exponential functions over an interval (t 1, t 2 ). Such representation is known as the exponential Fourier series representation. The coefficients in this series can be found by multiplying both sides by and integrating with respect to t over the interval. As a result of orthogonality, all terms on the right-hand side vanish except for m = n

2000Communication Fundamentals31 Representation of periodic signal by Fourier Series A periodic signal is such that It is assumed that the signal has finite energy over an interval (t 0, t 0 +T). We further assume that the energy content is constant over any interval of T seconds long. The power of the signal is, therefore, constant. The series representation will represent a periodic function over the infinite interval and the representation converges in a mean-square sense.

2000Communication Fundamentals32 The interval of integral to determine the Fourier series is taken over the complete period T. Example: Determine the Fourier series expression for the above waveform.

2000Communication Fundamentals33 To derive the trigonometric terms of the Fourier series we first consider n=1 and -1

2000Communication Fundamentals34 For n=1 Thus, the Fourier series term for  0 is

2000Communication Fundamentals35 Parseval’s Theorem For Power Signals The average power developed across a 1-Ohm resistance is where We may interchange the order of summation and integration Since the complex exponential functions are orthogonal over T, the integral will be zero except for m=n. For this case, the double summation reduces to a single summation giving --- Parseval’s Theorem

2000Communication Fundamentals36 Determine the average power of

2000Communication Fundamentals37 Transfer Function A system can be characterized in both time and frequency domains. In both approaches, linear superposition is assumed to add up the responses of the system for combinations of elemental functions. In this course we will focus on representation in the frequency domain. H(  ) is known as the frequency transfer function of the system, which, in general, can be expressed as |H(  )| is the magnitude response and  (  ) is the phase shift of the system

2000Communication Fundamentals38 The system response to a periodic signal is The output power can be determined by using Parseval’s theorem

2000Communication Fundamentals39 Example: Determine the output, g(t), of a linear time- invariant system whose input and frequency transfer function are as shown The Fourier representation for the input is

2000Communication Fundamentals40 The average input power is The output power is

2000Communication Fundamentals41 Harmonic Generation An important application of the Fourier series representation is in the measurement of the generation of harmonic content. A device with a nonlinear output-input gain characteristics can be used to accomplished this, e.g. If we let, then The nonlinear output-input characteristic has therefore resulted in the generation of a second-harmonic term. This device is known as the frequency doubler. Similarly, a third-order nonlinearity results in generation of third- harmonic content, etc.

2000Communication Fundamentals42 The presence presence of harmonic content in the output when only a single-frequency sinusoid is applied to the input represents distortion resulting from nonlinearities in the amplifier. A convenient way to measure this distortion is to take the ratio of the mean-square harmonic distortion terms to the mean-square of the first harmonic. This is known as the total harmonic distortion where a n and b n are the coefficients for the cosine and sine Fourier series components respectively

2000Communication Fundamentals43 The Fourier Spectrum It is the plot of the Fourier coefficients as a function of the frequency. In general, F n are complex-valued. To describe the coefficients will require 2 graphs, the magnitude spectrum and phase spectrum. Example: Sketch the amplitude spectrum and the magnitude and phase spectrum of From our previous analysis Thus The solutions are shown in the next figure

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2000Communication Fundamentals45 Example: Find the Fourier spectrum for the periodic function shown below: The Fourier coefficients are

2000Communication Fundamentals46 For, we have Thus, the exponential Fourier representation of the periodic gate function is

2000Communication Fundamentals47 It is important to note that: –the amplitude of the spectrum decrease as 1/T –the spacing between lines vary as

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2000Communication Fundamentals49 Numerical Computation of Fourier Coefficients Fourier series coefficients may be approximated numerically. The trigonometric Fourier coefficients are Approximating the integration we have where

2000Communication Fundamentals50 Example: Using 100 equally spaced sample points per period, compute the coefficients of the first 10 harmonic terms of the trigonometric Fourier series for the triangular waveform

2000Communication Fundamentals51 Thus,