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Appendix 1 Linear Systems Analysis Appendix 1 Linear Systems Analysis LINEAR SYSTEMS AND FILTER A linear system is given by the following relation: y(t) = f: h(t, 'O)x(r) d'O (A1-1) where x(t) is called the input, h(t, '0) the system impulse response function at t = '0, y(t) the output signal. If the system is time invariant, the output signal of the linear system is given by y(t) = f:oo h(t - 'O)x('O)d'O (A1-2) h(t) is called the impulse response. Because if the input is an impulse, then the output y(t) = f:oo h(t - '0)15('0) d'O = h(t) (A1-3) For a time invariant linear system, when the input is x(t - to), then the output is y(t - to), where to is the delay. The system is called causal if h(t) = 0, t<O (A 1-4) This means there is no response prior to the input. The system is called stable if bounded input produces a bounded output. This requires that the impulse function satisfies (A1-5) The time invariant system satisfies the linearity condition: If i = 1,2 549 550 APPENDIX 1 then (Al-6) where at and a2 are constants. Note that the output of a causal time invariant system is given by yet) = r", h(t - r)x(r) dr ~ h(t) ® x(t), (Al-7) convolution of x(t) with h(t). Taking the Fourier transform of the both sides of (Al-7), when they exist, we get Y(f) = H(f)X(f) (Al-8) where i2 j X(f) = 9"(x(t)) = J:", x(t)e- 7[ t dt i2 j H(f) = 9"(h(t)) = J:", h(t)e- 7[ t dt i2 j Y(f) = 9"(y(t)) = L"'", y(t)e- 7[ t dt Y(f), H(f), and X(f) are called the output, system, and input transfer func­ tion. In other words the system transfer function, the Fourier transform of the impulse function, is given by the relation Y(f) Fourier transform of output signal (Al-9) H(f) = X(f) = F·ouner transf orm 0 f·mput sIgna. I The time invariant linear system can also be given by the constant coefficient linear differential equation with a forcing term. The system is shown in Fig. (Al-l). X(t) H(t) y(n Output Input J I hIt) x(~ y(~ Fig. AI-l. Linear time invariant system. LINEAR SYSTEMS ANALYSIS 551 HILBERT <QUADRATURE) filTER H(f) = {-~, j: ~ j, f> 0 (A1-1O) (j=~) Taking the inverse Fourier transform of (A1-10), the filter impulse function 1 h(t) =- (A1-11) nt The output, known as the Hilbert transform is given by 1 y(t) = - (8) x(t) m = ~foo X(T) dt n -oot-T (Al-12) If x(t) = cos t, y(t) = sin t. The Hilbert transform shifts the phase and is there­ fore called quadrature filter. A signal z(t) is called analytic signal if z(t) = x(t) + ix(t), i=~ (Al-13) where x(t) is the Hilbert transform of x(t). If x(t) = sin t, then z(t) = exp [it]. SAMPLING Signals that are of discrete- type are called digital signals. Analog signals can be sampled, the sampled values of the signals form digital signals. It is possible to reconstruct the analog signal from its sampled values provided the signal is band limited and it is sampled at the Nyquist rate. A signal is called bandlimited of fx if its one sided spectrum is zero outside fx. The Nyquist sampling interval is defined as T, = 1/2fx, the Nyquist sampling rate is Is = 2fx, and Nyquist instants are (A1-14) For example, if x(t) = A cos 12m, then fx = 6 Hz, Nyquist sampling interval is 1/12 sec, and rate is 12 Hz. 552 APPENDIX 1 IDEAL INSTANTANEOUS SAMPLING THEOREM A band limited signal of bandwidth Ix can be exactly reconstructed from its sampled values uniformly spaced in time with period 1'. < 1/21x by passing it through an ideal low pass filter with bandwidth B where Ix < B < Is - Ix and is given by A ~ • 2B(t - n1'.) x(t) = 2B L... x(n1'.) sm ( ) (AI-15) n=-oo 2Bt-n1'. This formula is known also as interpolation or as cardinal series. z-TRANSFORM Let {Xn} be samples of a continuous signal. We define the z-translorm of Xn as 00 X(z) = L Xnz-n (AI-16) n~ -00 where z is a complex variable. The set of values of z for which the series converges to a finite value is called the domain or region of convergence. The z-transform X(z) is analytic in the region Rl < Izl < Rz whenever the right side of Eq. (AI-16) has a finite limit. If Xn = °when n < 0, then 00 X(z) = L Xnz-n (AI-17) n=O The z-transform of Xn is a series of negative power of z only. The z-transform exists if (AI-18) If z = R exp [i2nf], then X(z) exists if The series converges absolutely if (AI-19) LINEAR SYSTEMS ANALYSIS 553 The domain of convergence is outside of the circle of radius R I . If Xn = 0 when n > 0 then o 00 X(z) = L Xnz-n = L: X_mzm (Al-20) n=-oo m=O X(z) is a power series of positive power of z. The series converges absolutely if Izl < R z. The domain of convergence is the circle with center at the origin and radius R z. The transform of Xn in Eq. (Al-16) can be written as o 00 X(z) = L: Xnz-n + L: Xnz-n - Xo (Al-21) n=-oo n=O For the existance of X(z), the domain convergence is RI ~ Z ~ R z. When X(z) is a rational function of z, X(z) = P(z)/Q(z), where P(z), and Q(z) are polynomials in z. The roots of P(z) are called the zeros of X(z) and the roots of Q(z) are called the poles of X (z). When X(z) is a rational function of z, the domain of convergence is bounded by the minimum and maximum values of the poles. Properties of the z-transform 1. Linearity. If 00 L: Xnz-n = X(z), n=-oo and 00 L: Y,.z-n = Y(z), n=-oo then 00 L: (aXn + bYn)z-n = aX(z) + b Y(z), Rs < Izl < R6 (Al-22) n=-co where 554 APPENDIX 1 Rs = max[Rl,R3J R6 = min [R2 ,R4 J 2. Delay. If n= -00 then co L Xn_mz-n = z-m X (z), (Al-23) n=-oo 3. Conjugation. If co L Xnz-n = X(z), n= -00 then co L X:z-n = X*(z*), (Al-24) n=-X Here X: stands for complex conjugate of Xn and Xn is a complex sequence. 4. Convolution. If co L Xnz-n = X(z) n= -00 co L hnz-n = H(z) n= -00 and co Y" = L hn-kXk K=-co then Y(z) = H(z)X(z) (Al-25) The inverse of z-transform is given by LINEAR SYSTEMS ANALYSIS 555 Xn = ~ rX(Z)Zn-l dz (Al-26) 2mJc Where c is a closed path in the region of convergence in the counterclockwise direction. If X(z) is a rational function of Z, Xn can be evaluated by the partial fraction method or by the residue theorem Xn = ~ rX(z)zn-l dz 2mJc = { I [residues of X(z)zn-l at the poles inside c] for n ~ 0 (Al-27) - I [residues of X(z)zn-l the poles outside c] for n < 0 5. Parseval's theorem. 00 1 f dw I Xn Y,,* = -. X(w) Y*(l/w*)- (Al-28) n=-oo 2m c w 6. Complex convolution. If then U(z) = I00 unz-n = -.1 f X(w) Y (z)-- dw n=-oo 2m c w w (Al-29) where the region of convergence of X(z) is Rl < Izl < R2 and that of Y(z) is R3 < Izl < R 4 · A linear system is called bounded input bounded output (BIBO) stable, if the output is bounded for every bounded input. Let Xn be a bounded input sequence, hn be the impulse response sequence, and Y" be the output of the linear time-invariant discrete-time system. The output Y" is given by 00 Yn = I hkXn - k no:::::: -00 If the system is causal, then hk = 0 for k < 0, (Al-30) 556 APPENDIX 1 in this case, 00 Y" = L hkXn- k k=O If a linear, causal and time invariant system is stable, then the output is bounded IY"I<M when the input IXnl < L, Land M are constants. Since 00 Y" = L hkXn-k k=O then 00 I Y"I = L IhkllXn-kl k=O The causal system is stable if where C is a constant. On the other hand if then I Y"I is not bounded. We conclude that a linear time invariant and causal system is stable if and only if (Al-31) LINEAR SYSTEMS ANALYSIS 557 The z-transform of the causal linear time invariant system is 00 H(z) = L hnz-n n=O The region of convergence of H(z) is Izl > R2 because H(z) is a polynomial in negative powers of z. The system is causal if and only if the region of conver­ gence of H (z) is inside a circle of finite radius in the z-plane.
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