Signals and Systems: Introduction

Dependent variable Signals and Systems: Introduction What is a signal? Signals may describe a wide variety of physical phenomena. The information in a signal is contained in a pattern of variations of some form. A signal is represented mathematically as a function of one or more independent variables. y = f(x) Independent variable v A signal, where voltage (v) varies with time (t) t x CEN34: Signals and Systems; Ghulam Muhammad

.5 Amplitude, mv Examples of Signals - One dimensional signal, because there is only one independent variable, such as time. Speech Signal ECG (Electrocardiogram) Signal Amplitude -.5-2 3 4 Time (ms) Time (ms) CEN34: Signals and Systems; Ghulam Muhammad 2

Examples of Signals - 2 x x y y t Intensity of the image at location (x, y) can be expressed as I (x, y). As there are two independent variables (x and y), the image is a two dimensional signal. A video has three independent variables (x, y, and t (time)), therefore, it is a three dimensional signal. A video is a sequence of frames (images). CEN34: Signals and Systems; Ghulam Muhammad 3

Two Basic Types of Signals Continuous Signal A continuous-time (CT) signal is one that is present at all instants in time or space, such as oscillating voltage signal. Discrete-time Signal A discrete-time (DT) signal is only present at discrete points in time or space. For example closing stock market average is a signal that changes only at discrete points in time (at the close of each day). CEN34: Signals and Systems; Ghulam Muhammad 4

Discrete amplitude Continuous amplitude Continuous amplitude Continuous, discrete-time, & Digital Signals Continuous Signal Discretetime Signal Continuous time Discrete time Digital Signal Discrete time You will learn it in CEN 352. CEN34: Signals and Systems; Ghulam Muhammad 5

Notation of Continuous and discretetime Signals To distinguish between continuous-time and discrete-time signals, we will use The symbol t to denote the continuous-time independent variable and n to denote the discrete-time independent variable. We will enclose the independent variable in parentheses (. ) and for discrete-time signals, we will use brackets [. ] x(t) x[n] t - 2 3 4 n CEN34: Signals and Systems; Ghulam Muhammad 6

Continuous and discrete-time Signals Independent variable t We can convert a continuous signal into a discrete-time signal by sampling. Sampling period Independent variable n n is always an integer. CEN34: Signals and Systems; Ghulam Muhammad 7

Systems A system is an abstraction of anything that takes an input signal, operates on it, and produces an output signal. A system generally establishes a relationship between its input and its output. Examples could be car, camera, etc. Systems that operate on continuous-time signal are known as continuous-time (CT) systems. Systems that operate on discrete-time signals are known as discrete-time (DT) systems. Input Signal system Output Signal CEN34: Signals and Systems; Ghulam Muhammad 8

Examples of Systems An RLC circuit Automatic speech recognition (ASR) system ASR System I am Mr. Red Courtesy of Prof. Alan S. Willsky What is the input signal? x(t) (the D.C. source) What is the output signal? What is the system? y(t) (the signal across capacitor) The whole RLC network Imaging system Image CEN34: Signals and Systems; Ghulam Muhammad 9

Drill -. Most of the signals in this physical world is. (CT signals / DT signals). Choose the right one. 2. Mention four systems other than those mentioned in the slides. 3. Mention three signals other than those mentioned in the slides. 4. How can we convert a CT signal into a DT signal? 5. Can a system have multiple inputs and multiple outputs? 6. What do you mean by time-domain signal and spatial-domain signal? CEN34: Signals and Systems; Ghulam Muhammad

MATLAB Matlab is a software tool for computation in science and engineering. Developed, published and trademarked by The MathWorks, Inc. Originally developed as a Matrix Laboratory but now used in applications in almost all areas of science and engineering. It has a rich collection of tool boxes covering basic mathematics, graphics, differential equations, electric/electronic circuits, partial differential equations, simulation problems, control systems, signal processing, image processing, statistics, symbolic computations, etc. http://www.mathworks.com/help/pdf_doc/matlab/getstart.pdf http://www.mathworks.com/academia/student_center/tutorials/launc hpad.html CEN34: Signals and Systems; Ghulam Muhammad

..2 Signal Power and Energy Continuous-time (CT) signal The total energy over the time internal t t t 2 in a continuous-time signal x(t) is defined as t 2 2 x ( t ) dt t where x denotes the magnitude of the (possibly complex) number x. The time averaged power is given by t 2 t Over an infinite time interval, i.e., for < t < + Total energy: Total averaged power: T E lim T න T ሻ x(t T P lim T 2T න T CEN34: Signals and Systems; Ghulam Muhammad 2 න t 2 dt = න ሻ x(t 2 dt t 2 ሻ x(t ሻ x(t 2 dt 2 dt

..2 Signal Power and Energy Discrete-time (DT) signal The total energy in a discrete-time signal x[n] over the time interval n n n 2 is defined as n 2 ] n=n x[n 2 The average power over the interval in this case is given by n 2 n + Over an infinite time interval, i.e., for < t < + Total energy: Total averaged power: +N E lim N n= N ] x[n 2 = +N P lim N 2N + n= N + n= x[n] 2 x[n] 2 n 2 ] n=n CEN34: Signals and Systems; Ghulam Muhammad 3 x[n 2

Three Important Cases Case : Signals with finite total energy, i.e., E < : Such a signal must have zero average power. For example, in continuous case, if E <, then E P = lim T 2T = An example of a finite-energy signal is a signal that takes on the value of for t and otherwise. In this case, E = and P =. Case 2: Signals with finite average power, i.e., P < : For example, consider the constant signal where x[n] = 4. This signal has infinite energy, as +N E = lim N n= N +N ] x[n 2 = lim N n= N 4 2 = + 6 + 6 + 6 CEN34: Signals and Systems; Ghulam Muhammad 4

Three Important Cases - continued However, the total average power is finite, +N P lim N 2N + n= N +N x n 2 = lim N 2N + n= N 4 2 6 = lim σ +N 6 2N+ N 2N+ n= N = lim N 2N+ = lim N 6 = 6 Case 3: Signals with neither E nor P finite: A simple example of such a case could be x(tሻ = t. In this case both E and P are infinite CEN34: Signals and Systems; Ghulam Muhammad 5

Some Frequently Used Signals CEN34: Signals and Systems; Ghulam Muhammad 6

Example: Power and Energy Problem : Find P and E for the signal, x t e u t ( ) 2t ( ) Solution: 2 2t 2 2t 2 E x ( t ) dt e u ( t ) dt e dt 4t = e dt 4() 4( ) e e 4 4 4 P is zero, because E < CEN34: Signals and Systems; Ghulam Muhammad 7

.2 Transformations of the Independent Variable The transformation of a signal is one of the central concepts in the field of signals and systems. We will focus on a very limited but important class of signal transformations that involves the modifications of the independent variable, i.e., the time axis. (A) Time Shift The original and the shifted signals are identical in shape, but are displaced or shifted along the time-axis with respect to each other. Signals could be termed as delayed or advanced in this case. CEN34: Signals and Systems; Ghulam Muhammad 8

Time Shift Signals Continuous time Discrete time Original Delayed Advance CEN34: Signals and Systems; Ghulam Muhammad 9

Time Shift - continued Such signals arise in applications such as radar, sonar and seismic signal processing. Several receivers placed at different locations receive the time shifted signals due to the transmission time they take while passing through a medium (air, water or rock etc.). CEN34: Signals and Systems; Ghulam Muhammad 2

Time Reversal (Reflection) In this case, the original signal is reflected about the time =. For example, if the original signal is some audio recording, then the time reversed signal would be the audio recording played backward. Signals Continuous time Discrete time Original Time reversed CEN34: Signals and Systems; Ghulam Muhammad 2

Time Scaling In this case, if the original signal is x(tሻ, the time variable is multiplied with a constant to get a time-scaled signal, e.g., x(2tሻ, x 5t, or x( tτ2ሻ. If we think of the signal x(tሻ as audio recording, then x(2tሻ is the audio recording played at twice the speed and x( tτ2ሻ is the recording played at half of the speed. Signals Continuous time Original Stretching Compressing CEN34: Signals and Systems; Ghulam Muhammad 22

General Case of the Transformation of the Independent Variable A general case for the transformation of independent variable is the one in which for the original signal x(tሻ is changed to the form x(αt + βሻ, where α and β are given numbers. It has the following effects on the original signal: The general shape of the signal is preserved. The signal is linearly stretched if α <. The signal is linearly compressed if α >. The signal is delayed (shifted in time) if β <. The signal is advanced (shifted in time) if β >. The signal is reversed in time (reflected) if α <. CEN34: Signals and Systems; Ghulam Muhammad 23

Example: Time Shift: () x t = 2 t if if if if t < t < t < 2 t 2 The signal x t + can be obtained by shifting the given signal to the left by one unit CEN34: Signals and Systems; Ghulam Muhammad 24

Example: Time Shift: (2) The signal x t + can be obtained using the mathematical definition or figure of the original signal x t. If we use the mathematical definition, then making the following table could be useful. t t + x t + 2 3.. 5 2.5 2.. 5.5.5.. 5.5.. 5.5 2. 2. 5.5 3 2. First plot x(t+), then reflect. CEN34: Signals and Systems; Ghulam Muhammad 25

MATLAB Drill - In MATLAB, the original signal can be written as an inline function. This function can then be used to plot the original signal, the shifted signal and the timereversed signal using the following MATLAB code. >>g = inline(' ((t>=)&(t<)) + (2-t).*((t>=) & (t<2))','t'); >>t = -3:.:3; >>subplot(3,,), plot(t, g(t)), axis([-3 3 -..]), title('original Signal') >>subplot(3,,2), plot(t, g(t+)), axis([-3 3 -..]), title('time-shifted Signal') >>subplot(3,,3),plot(t, g(-t+)),axis([-3 3 -..]), title('time-reversed Signal') CEN34: Signals and Systems; Ghulam Muhammad 26

MATLAB Drill : continued Original Signal.5-3 -2-2 3 Time-Shifted Signal.5-3 -2-2 3 Time-Reversed Signal.5-3 -2-2 3 CEN34: Signals and Systems; Ghulam Muhammad 27

Example: Time Compression: () x t = 2 t if if if if t < t < t < 2 t 2 Find x 3 t 2 x(αt + βሻ; α >, so linear compression by a factor of / (3/2) = 2/3 Compressed Signal 2/3 = 2/3 2 2/3 = 4/3 t CEN34: Signals and Systems; Ghulam Muhammad 28

Example: Time Compression: (2) Find x 3 t 2 Compressed by a factor of 2/3, and shift left by Shift left by Compressed by 2/3 - t - 2/3 =- 2/3 Final Signal 2/3 = 2/3 CEN34: Signals and Systems; Ghulam Muhammad 29 t

.2.2 Periodic Signals A periodic continuous-time signal x(tሻ is defined as x t = x(t + Tሻ where T is a positive number called the period. A typical example is that of a sinusoidal signal x t = sin(tሻ for < t < +. For the above signal, the period is T = 2. It can be noticed that for any time t: ሻ sin t + 2π = sin(t ሻ sin t + m2π = sin(t where m is a positive number. CEN34: Signals and Systems; Ghulam Muhammad 3

Periodic Signals - continued The fundamental period T of x(tሻ is the smallest positive value of T for which the equation x t = x(t + Tሻ holds. A discrete-time signal x[n] is periodic with period N, where Nis a positive integer, if it is unchanged by a time-shift of N, i.e., if ] x n = x[n + N for all values of n. Period, N = 4 The fundamental period N of x[n] is the smallest positive value of N for which the equation x[n] = x[n + N ] holds. CEN34: Signals and Systems; Ghulam Muhammad 3

Periodic Signals - Example x() t cos( t) if t sin( t) if t Since, cos(2+t) = cos(t) and sin(2+t) = sin(t), considering t < and t separately, the signal repeats itself in every interval of 2. But, if we look at the following figure of x(t), we find there is a discontinuity at t =, which does not occur at any other time. Therefore, x(t) is not periodic. CEN34: Signals and Systems; Ghulam Muhammad 32

.2.3 Even and Odd Signals Even Signals A signal x(tሻ or x[n] is defined as an even signal if it is identical to its time-reversed counterpart, i.e., with its reflection about the origin. Even continuous-time Signal Even Discrete-time Signal x t = x(tሻ x[ n] = x[n] CEN34: Signals and Systems; Ghulam Muhammad 33

Odd Signals A signal x(tሻ or x[n] is defined as an odd signal if, Odd continuous-time Signal Odd Discrete-time Signal x t = x(tሻ x n = x[n] As a special case, the odd signal must be zero at t = or n =. CEN34: Signals and Systems; Ghulam Muhammad 34

Decomposing a Signal into Even and Odd Parts An important fact is that any signal (continuous-time or discrete-time) can be broken into a sum of two signals: even and odd. Signal Component Mathematical Form Continuous-time Signal x(tሻ Even Part E v x t = 2 Odd Part O d x t = 2 x t + x( tሻ x t x( tሻ Discrete-time Signal x[n] Even Part Odd Part E v x[n] = 2 O d x[n] = 2 x[n] + x[ n] x[n] x[ n] CEN34: Signals and Systems; Ghulam Muhammad 35

Decomposing a Signal into Even and Odd Parts Example = + CEN34: Signals and Systems; Ghulam Muhammad 36

.3 Exponential and Sinusoidal Signals Continuous-Time Complex Exponential and Sinusoidal Signals A continuous-time complex signal x(tሻ can be written as x t = Ce at where C and a are, in general, complex numbers. Real Exponential Signals In this case both C and a are real numbers, and x(tሻ is called a real exponential. Continuous-time Real Exponential with a> Continuous-time Real Exponential with a< x(t)=ce at, C>, a> x(t)=ce at, C>, a< C C t CEN34: Signals and Systems; Ghulam Muhammad 37 t

Periodic Complex Exponential and Sinusoidal Signals Now we consider the case of complex exponentials where a is purely imaginary. More, specifically, we consider: x t = e jω t An important property of this signal is that it is periodic. x t x t T e e e e e jt j ( t T ) jt jt jt ( ) ( ) This equation can be true,. If, ω =, then x t =, which is periodic for any value of T. 2. If, ω, then the fundamental period T of x t, i.e. the smallest value of T for which the above equation holds, is T = 2π ω CEN34: Signals and Systems; Ghulam Muhammad 38

Periodic Signals Replacing the value of Twith this T, and using Euler s formula, that is, We get e jω T = cos(ω Tሻ + jsin(ω Tሻ e jω T = cos 2π + j sin 2π = + j = Therefore, the signal x t is a periodic signal. Similarly, the signal x t = e jω t has the same fundamental period. Sinusoidal Signal: x t = A co s( ω t + φሻ Continuous-Time Sinusoidal Signal x(t)=a cos( t+) T A A cos() 2 t CEN34: Signals and Systems; Ghulam Muhammad 39

Sinusoid Signals A cos ω t + φ = A ej ω t+φ + j(ω e t+φሻ 2 = A 2 ejφ e jω t + A 2 e jφ e jω t A cos ω t + φ = A Re e A sin ω t + φ = A Im e j(ω t+φሻ j(ω t+φሻ The fundamental period T of a continuous-time sinusoidal or a periodic complex exponential signal, is inversely proportional to the ω, which is called the fundamental frequency. T = 2π ω CEN34: Signals and Systems; Ghulam Muhammad 4

Fundamental Period and Frequency If we decrease the value of the magnitude of ω, we slow down the rate of oscillations and hence increase the period T. Alternatively, if we increase the value of the magnitude of ω, we increase the rate of oscillations and hence decrease the period T. 2 3 T T T 2 3 CEN34: Signals and Systems; Ghulam Muhammad 4

Energy & Power of Sinusoid / Complex Exp Signals Over the one fundamental period T of a continuous-time sinusoidal or a periodic complex exponential signal, the signal energy and power can be determined as: E period = න T T e jω t 2 dt = න dt = T P period = T න e jω t 2 dt = T න dt = T = T T T As there are an infinite number of periods as t ranges from to +, the total energy integrated over all time is infinite. The total average power is however remains, as by definition, T P = lim T 2T න T e jω t 2 dt = lim T 2T 2T = CEN34: Signals and Systems; Ghulam Muhammad 42

Harmonics of a Periodic Complex Exponential We have noted that, e jωt = which implies that ωt is a multiple of 2π, i.e., ωt = 2πk where k =, ±, ±2, This shows that ω must be an integer multiple of ω, i.e., the fundamental frequency. We can therefore, write φ k t = e jkω t where k =, ±, ±2, This is called the k-harmonic of the complex exponential signal. CEN34: Signals and Systems; Ghulam Muhammad 43

Expressing Two Complex Exponentials into a Product of One Complex Exp. & One Sinusoidal x j2t ( t) e e j3t x( t) e j2.5t j.5t j.5t j2.5t e e 2e cos(.5t) The magnitude of x(t) is: x( t) 2 cos(.5t) Full-wave rectified sinusoid. CEN34: Signals and Systems; Ghulam Muhammad 44

General Complex Exponential Signals The general complex exponential signals are of the form x t = Ce at Where both C and a are complex numbers. Let us represent them as C = C e jθ a = r + jω Polar form Cartesian form x t = Ce at = C e jθ e (r+jω ሻt = C e rt e j(ω t+θሻ x t = Ce at = C e rt cos ω t + θ + j C e rt sin ω t + θ. For r =, the real and imaginary parts of a complex exponential are sinusoidal. 2. For r >, they correspond to sinusoidal signals multiplied with growing exponential. 3. For r <, they correspond to sinusoidal signals multiplied with decreasing exponentials. CEN34: Signals and Systems; Ghulam Muhammad 45

Example: General Complex Exponential Signals Sinusoid with growing exponential x(t)= C e rt cos( t+) Sinusoid with decaying exponential x(t)= C e -rt cos( t+) r > r < t t Damped sinusoid May occur in an RLC network due to resistors CEN34: Signals and Systems; Ghulam Muhammad 46

.3.2 Discrete-Time Complex Exponential and Sinusoidal Signals A discrete-time complex exponential signal or sequence x[n] can be written as x[n] = Cα n where C and α are, in general, complex numbers. This could also be written as Real Exponential Signals x[n] = Ce βn where α = e β In this case both C and α are real numbers, and x[n] is called a real exponential. USAGE: Real-valued discrete-time exponentials are often used to describe population growth as a function of generation, and total return on investment as a function of day, month, a quarter. CEN34: Signals and Systems; Ghulam Muhammad 47

Example: Real Exponential Signals x[n] = C n (a) > ; (b) < < ; (c) - < < ; (d) < - What will happen if (i) =, and (ii) = -? CEN34: Signals and Systems; Ghulam Muhammad 48

Discrete-Time Sinusoid Signals x[n] = e jω n = cos ω n + j si n( ω n൯ Therefore, a discrete-time sinusoid signal can be written as: A cos ω n + φ = A ej ωn+φ j(ω ሻ + e n+φ 2 = A 2 ejφ e jω n + A 2 e jφ e jω n Using real and imaginary parts, we find: j(ω A cos ω n + φ = A Re e n+φሻ A sin(ω n + φሻ = A Im e j(ω n+φሻ Both the shaded signals have infinite total energy, but finite average power. For example, for every sample, e jω n 2 =, so it contributes to the total energy, making it infinite; however, per point time, the average power is. CEN34: Signals and Systems; Ghulam Muhammad 49

Example: Discrete-Time Sinusoid Signals CEN34: Signals and Systems; Ghulam Muhammad 5

Discrete-Time Complex Exponential Signals The general discrete-time complex exponential signals are of the form x[n] = Cα n where both C and α are complex numbers. Let us represent them as C = C e jθ α = α e jω Polar form x[n] = Cα n = C e jθ α n e jω n = C α n e Using Euler s formula, it can be written as j(ω n+θሻ x[n] = Cα n = C α n cos ω n + θ + j C α n sin ω n + θ. For α =, the real and imaginary parts of a complex exponential are sinusoidal. 2. For α >, they correspond to sinusoidal signals / sequences multiplied with growing exponential. 3. For α <, they correspond to sinusoidal signals / sequences multiplied with decreasing exponentials. CEN34: Signals and Systems; Ghulam Muhammad 5

Growing Sinusoidal Signal Discrete-Time Complex Exponential Signals Growing Sinusoidal Signal n n Decaying Sinusoidal Signal Decaying Sinusoidal Signal n CEN34: Signals and Systems; Ghulam Muhammad 52 n

Discrete-Time Complex Exponential Signals There are many similarities between continuous-time and discrete-time signals. But also there are many important differences. One of them is related with the discrete-time exponential signal e jω n The following properties were found with regard to the continuous-time exponential signal e jω t :. The larger the magnitude of ω, the higher is the rate of oscillations in the signal; 2. e jω t is periodic for any value of ω. To see the difference for the first property, consider the discrete-time complex exponential: e j(ω +2πሻn = e j2πn e jω n = e jω n This shows that the exponential at ω + 2π is the same as that at frequency ω CEN34: Signals and Systems; Ghulam Muhammad 53

Discrete-Time Complex Exponential Signals In case of continuous-time exponential, the signals e jω t are all distinct for distinct values of ω. In discrete-time, these signals are not distinct. In fact, the signal with frequency ω is identical to signals with frequencies ω ± 2π, ω ± 4π and so on. Therefore, in considering discrete-time complex exponentials, we need only consider a frequency interval of size 2π. The most commonly used 2π intervals are ω 2π or the interval π ω π. As ω is gradually increased, the rate of oscillations in the discrete-time signal does not keep on increasing. If ω is increased from to 2π, the rate of oscillations first increase and then decreases. Note in particular that for ω = π or for any odd multiple of π, e jπn = e jπ n = n so that the signal oscillates rapidly, changing sign at each point in time. CEN34: Signals and Systems; Ghulam Muhammad 54

Discrete-Time Complex Exponential Signals x[n] = cos(*n) x[n] = cos( n/8) x[n] = cos( n/4).5 2 3-2 3-2 3 x[n] = cos( n/2) - 2 3 x[n] = cos( n) - 2 3 x[n] = cos(3 n/2) - 2 3 Start decreasing from here x[n] = cos(7 n/4) x[n] = cos(5 n/8) x[n] = cos(2 n).5-2 3-2 3 2 3 CEN34: Signals and Systems; Ghulam Muhammad 55

Periodicity of Discrete-Time Complex Exponential Signals ሻ ejω (n+n = e jω n e jωn = It is true if ω N is a multiple of 2π ω ω N = 2πm 2π = m N It means that the discrete-time signal e jω n is periodic only when ω 2π e jω t e jω n is a rational number. Distinct signals for distinct values of ω. Identical signals for values of ω separated by multiples of 2π. Periodic for any choice of ω. Periodic only if ω = 2πmΤN for some integer N > and m. Fundamental frequency ω. Fundamental frequency ω /m. Fundamental period Fundamental period ω = : undefined ω = : undefined ω : 2π ω ω : m 2π ω CEN34: Signals and Systems; Ghulam Muhammad 56

Periodicity; Workout () Find fundamental period of the signal: The first term 2 / 3 2 2 N m m 2 / 3 N 3 N 3 m x [ n] e e The second term 3 / 4 j (2 /3) n j (3 /4) n 2 2 N m m 3 / 4 N 8 N 8 m 3 LCM(3, 8) = 24 Therefore, the fundamental period = 24 CEN34: Signals and Systems; Ghulam Muhammad 57

Cartesian to Polar & Vice Versa; Workout (2) j e 2 (/ 2) cos j sin j e 2 (/ 2) cos( ) j sin( ) (/ 2)( j ) (/ 2)( j ) (/ 2) j () (/ 2) j () 3 j j Ce C cos Cj sin 3 j () C 3 sin / 2 3j 3e j /2..2 CEN34: Signals and Systems; Ghulam Muhammad 58

.4 (a) Workout (3) Let, x[n] be a signal with x[n] = for n < -2 and n > 4. For a signal x[n-3], determine the value of n for which it is guaranteed to be zero. x[n-3] means shifting the signal towards right by 3 samples..8 n < -2 n+3 < -2+3 (=) Amplitude.6.4 n > 4 n+ 3 > 4+3 (=7).2-4 -2 2 4 6 8 t The shifted signal will be zero for n < and n > 7. CEN34: Signals and Systems; Ghulam Muhammad 59

.5 (a) Workout (4) Let x(t) be a signal with x(t) = for t < 3. For the signal x( t), determine the value of t for which it is guaranteed to be zero. x(t) x(-t) Amplitude.8.6.4 Amplitude.8.6.4 Left shift by and then reflect.2.2-5 5 t -4-2 2 4 6 t For t > -2, the signal is zero. CEN34: Signals and Systems; Ghulam Muhammad 6

.7 (a) Workout (5) For a signal x[n] = u[n] u[n-4], determine the values of the independent variable at which the even part of the signal is guaranteed to be zero. Amplitude Amplitude.8.6.4.2.5 EVEN{ x[n]} =.5 (x[n] + x[-n]) =.5 (u[n] u[n-4] + u[-n] u[-n-4]) u[n] Amplitude -5 5 t u[-n] - -5 5 t Amplitude.5 Amplitude Amplitude.8.6.4.8.6.2-5 5 t -5 5.8 t.5 u[n-4] u[-n-4] u[n] u[n-4] u[-n] u[-n-4].4 - -5 5.2 t - -5 5 t -4-2 t 2 4 6 CEN34: Signals and Systems; Ghulam Muhammad 6 Amplitude.6.4.2.5 (x[n] + x[-n]) Zero for n > 3 and n <-3

.7 (b) Workout (6) For a signal x(t) = sin(.5t), determine the values of the independent variable at which the even part of the signal is guaranteed to be zero..5 Amplitude -.5 - -3-2 - 2 3 t It is always an odd signal, so the even part is zero for all values of t. CEN34: Signals and Systems; Ghulam Muhammad 62

.8 (a) Workout (7) Express the real part of the signal, x(t) = -2, in the form Ae -at cos(t+), where A, a,, and are real numbers with A >, and - < <. x(t) = A e -at cos(t + ) = -2 = 2 (-) = 2 e -t cos(t + ) A = 2, a =, =, and = j /4 The above problem when the signal is x ( t ) 2e cos3t 2 x t e t j t 4 4 Real part = 2 cos cos3t 2 2 cos 3t cos 3t 4 2 j /4 ( ) 2 cos 3 2 2 cos sin cos 3 2 t e cos(3t ) A =, a =, = 3, and = CEN34: Signals and Systems; Ghulam Muhammad 63

.9 (a) Workout (8) If the signal x(t) is periodic, find the fundamental period. x () t je t /2 jt j (cost j sin t ) j cost sint j sin t / 2 cos t / 2 e Fundamental period: T 2 2 5 CEN34: Signals and Systems; Ghulam Muhammad 64

. Workout (9) If the signal x(t) is periodic, find the fundamental period. x ( t ) 2 cos(t ) sin(4t ) T 2 T 5 2 4 2 Fundamental period: LCM ( /5, /2) = LCM (, ) / HCF (5, 2) = / = a c LCM, LCM ( a, c) / HCF ( b, d ) b d CEN34: Signals and Systems; Ghulam Muhammad 65

. Workout () Determine the fundamental period of the following signal x[n]. x [ n] e e j 4 n /7 j 2 n /5 N N 2 m (first part) 2 N (second part) m m (7 / 2) 7 4 / 7 2 N (third part) m m (5 / 2) 5 2 / 5 N LCM (, 7,5) 35 CEN34: Signals and Systems; Ghulam Muhammad 66

Acknowledgement The slides are prepared based on the following textbook: Alan V. Oppenheim, Alan S. Willsky, with S. Hamid Nawab, Signals & Systems, 2 nd Edition, Prentice-Hall, Inc., 997. Special thanks to Prof. Anwar M. Mirza, former faculty member, College of Computer and Information Sciences, King Saud University Dr. Abdul Wadood Abdul Waheed, faculty member, College of Computer and Information Sciences, King Saud University CEN34: Signals and Systems; Ghulam Muhammad 67