Title signals-notes.pdf ✅

Signals and Systems HT Last updated: August 10, 2024 Contents Prerequisite math 1 Introduction 1 Classification of signals 2 3.1 Even and odd signals . . . . . . . . . 2 3.2 Periodic and nonperiodic signals . . 3 3.3 Causal and noncausal signals . . . . 4 Basic time signals 4 4.1 Unit step signal . . . . . . . . . . . . 4 4.2 Unit impulse function . . . . . . . . 4 4.3 Ramp signal . . . . . . . . . . . . . . 5 4.4 Exponential signals . . . . . . . . . . 5 Operations on signals 5 5.1 Time shifting . . . . . . . . . . . . . . 5 5.2 Time reversal . . . . . . . . . . . . . 6 5.3 Addition of signals . . . . . . . . . . 6 5.4 Time scaling . . . . . . . . . . . . . . 6 5.5 Amplitude transformations . . . . . 7 Energy and power of signals 7 6.1 Energy signals . . . . . . . . . . . . . 7 6.2 Power signals . . . . . . . . . . . . . 7 Analogy of signals and vectors 8 Systems 8 8.1 Time variance . . . . . . . . . . . . . 8 8.2 Linearity . . . . . . . . . . . . . . . . 9 8.3 Static/dynamic . . . . . . . . . . . . 9 8.4 Causality . . . . . . . . . . . . . . . . 10 8.5 Stability . . . . . . . . . . . . . . . . . 10 Convolution integral 10 9.1 Linear time invariant system (LTI) . 10 9.2 Properties of convolution . . . . . . 11 9.3 Step response . . . . . . . . . . . . . 11 9.4 Graphical procedure . . . . . . . . . 12 9.5 Discrete time convolution . . . . . . 12 1 Prerequisite math The following are some important trigonometric formulae: sin(A + B) = sin A cos B + cos A sin B sin(A − B) = sin A cos B − cos A sin B cos(A + B) = cos A cos B − sin A sin B cos(A − B) = cos A cos B + sin A sin B Some identities are: sin2 A + cos2 A = 1 1 + tan2 A = sec2 A 1 + cot2 A = csc2 A 1 − cos 2A = 2 sin2 A 1 + cos 2A = 2 cos2 A 1 − sin 2A = (cos A − sin A) 2 1 + sin 2A = (cos A + sin A) 2 sin 2A = 2 sin A cos A cos 2A = 1 − 2 sin2 A = cos2 A − sin2 A = 2 cos2 A − 1 2 Introduction A signal is a function that carries some informa- tion, or some physical quantity that varies with time or space (or any other independent variable). If the signal is a function of one independent variable, it can be called a one dimensional signal. f(t) = at, −1 < t < 1 If it is a function of more than one independent variable, it is called a multi dimensional signal. For example: f(t,s) = at2 + bs2 , −1 < t < 1, −1 < s < 1 A system is an interconnection of devices, operat- ing on an input signal x(t) or excitation, and giv- ing an output signal y(t) or response. i/p signal x(t) → System → o/p signal y(t) 1

Analog Circuits 3.2 Periodic and nonperiodic signals Any signal x(t) can be represented as the sum of even and odd signals as: x(t) = xe(t) + xo(t) x(t) = x(t) + x(−t) 2 + x(t) − x(−t) 2 where the first fraction is the even component, and the second fraction is the odd component. Note that the product of two even signals or of two odd signals is an even signal and that the product of an even signal and an odd signal is an odd signal. Example 3.1. Find the odd and even components of e jt . The even component can be written as 1 2 e jt + e −jt or cos t and the odd component as 1 2 e jt − e −jt or j sin t. Example 3.2. Find the even and odd component of cos(ω0t + π 3 ). The even component can be written as 1 2 cos(ω0t + π 3 ) + cos(ω0(−t) + π 3 . This is of the identity 2 cos A+B 2 cos A−B 2 . Therefore, cos π 3 cos ω0t = 1 2 cos ωt. Similarly, the odd component can also be found. 3.2 Periodic and nonperiodic signals A periodic continuous time signal satisfies: x(t) = x(t + nT) where n is an integer and T is the period of the signal. Thus, it repeats itself every T seconds. T0 is the fundamental period of x(t), and is the smallest positive value of T for which the period- icity holds. f is the fundamental frequency and is equivalent to 1 T Hz. ω0 is the fundamental angular frequency equivalent to 2π T = 2π f . For a discrete time signal or sequence x[n + N] = x[n] The fundamental period N0 of x[n] is the smallest positive integer N for which the periodicity holds. Any sequence which is not periodic is called a nonperiodic sequence. Note that a sequence obtained by uniform sam- pling of a periodic continuous-time signal may not be periodic. Note also that the sum of two continuous-time periodic signals may not be pe- riodic, but that the sum of two periodic sequences is always periodic. Example 3.3. Sketch a continuous time signal x(2) = 2 sin πt in the interval 0 ≤ t ≤ 2. Sample this continuous time signal with sampling period T = 0.2 sec, and sketch discrete time signal. From x(t), we see amplitude of the signal is 2, and 2π f = 2π T = π, which means the time period is T = 2 seconds. Example 3.4. x(t) = e j3πt . Find the fundamental period. cos 3πt + j sin 3πt is the expansion. cos(3πt + 3πT) + j(sin(3πt + 3πT)) has to be equal to the expression to be periodic. Equating each separately, we obtain 3πT = 2nπ, or T = 2n 3 . This means T0 = 2 3 Example 3.5. Show cos(ωt + θ) is periodic. For periodicity, cos(ωt + ωT + θ) = cos(ωt + θ) Therefore, 2nπ = ωT, or T = 2nπ ω . The fundamen- tal period is T0 = 2π ω Example 3.6. x(t) = cos 15t. Find the value of sampling interval Ts such that x[n] is a periodic sequence. Find the fundamental period of x[n] if Ts = 0.1π seconds. We have T0 = 2π ω = 2π 15 For periodicity: Ts T0 = m N Example 3.7. f(t) = cos 2t − π 3 2 . Find period- icity and fundamental period. cos2 (2t − π 3 ) = 1+cos 2(2t− π 3 ) 2 For periodicity, 1+cos 2(2t− π 3 ) 2 = 1+cos 2(2t− π 3 +T) 2 The period of cos x is π radians, therefore T = nπ and T0 = π. Example 3.8. We have the function ∑ ∞ k = −∞ h e −(t−2k)u(t − 2k) − e −(t−2k)u(t − 2k − 1) i . Check periodicity and find T0. Example 3.9. x[n] = (−1) n Example 3.10. f(t) = cos t + sin √ t Example 3.11. f(t) = cos nπ 3 + sin nπ 4 . Find periodicity and fundamental period. 3