Common use of Solution Clause in Contracts

Solution. M1 = 2, M2 = 4, M3 = 4. F1(ω) = H(z) F2(ω) = G(z)H(z2) F3(ω) = G(z)G(z2) Discrete-Time Fourier Transform : X(ω) = n=Σ−∞ x[n]e−jωn ∞ ∫ Inverse Discrete-Time Fourier Transform : x[n] = 1 2π X(ω)ejωt dω . 2π x[n] X(ω) condition anu[n] 1 1 − ae−jω (n + 1)anu[n] 1 (1 − ae−jω )2 (n + r − 1)! anu[n] 1 |a| < 1 |a| < 1 |a| < 1 n!(r − 1)! (1 − ae−jω )r δ[n] 1 δ[n − n0] e−jωn0 ∞ x[n] = 1 2π k=Σ−∞ δ(ω − 2πk) ∞ 1 u[n] + 1 − e−jω ∞ k=Σ−∞ πδ(ω − 2πk) ejω0n 2π k=Σ−∞ ∞ δ(ω − ω0 − 2πk) cos(ω0n) π k=Σ−∞ ∞ {δ(ω − ω0 − 2πk) + δ(ω + ω0 − 2πk)} π j sin(ω n) Σ {δ(ω − ω − 2πk) − δ(ω + ω − 2πk)} ∞ k=Σ−∞ ∞ N k=−∞ 2π δ[n − kN ] N k=Σ−∞ δ ω − 2πk x[n] = ,,, 1 , |n| ≤ N , 0 , |n| > N sin(ω(N + 1/2)) sin(ω/2) sin(Wn) W Wn X(ω) = 1 , 0 ≤ |ω| ≤ W , = sinc πn π π , 0 , W < |ω| ≤ π X(ω) is periodic with period 2π x[n] D←T→F T X(ω) and y[n] D←T→F T Y (ω) Property Time domain DTFT domain Linearity Ax[n] + By[n] AX(ω) + BY (ω) Time Shifting x[n − n0] X(ω)e−jωn0 Frequency Shifting x[n]ejω0n X(ω − ω0) Conjugation x∗[n] X∗(−ω) Time Reversal x[−n] X(−ω) ∫ Convolution x[n] ∗ y[n] X(ω)Y (ω) Multiplication x[n]y[n] 1 2π − X(θ)Y (ω θ)dθ 2π Differencing in Time x[n] − x[n − 1] (1 − e−jω)X(ω) 1− −jωe Accumulation Σk∞=−∞ x[k] 1 + πX(0) Σ∞k=−∞ δ(ω − 2πk) dX(ω) Frequency Differentiation nx[n] j dω 2π 2π Parseval’s Relation for Aperiodic Signals Σ∞k=−∞ |x[k]|2 1 ∫ |X(ω)|2dω Z-Transform : X(z) = n=Σ−∞ x[n]z−n ∞ I Inverse Z-Transform : x[n] = 1 2πj X(z)zn−1 dz . C x[n] X(ω) ROC anu[n] 1 1 − az−1 |z| > |a| n −a u[−n − 1] 1 − az−1 |z| < |a| nanu[n] n az−1 (1 − az−1)2 |z| > |a| az−1 −na u[−n − 1] (1 − az−1)2 |z| < |a| δ[n] 1 All z δ[n − n0] z−n0 All z u[n] cos(ω0n)u[n] sin(ω0n)u[n] n 1 − z−1 |z| > 1 1 − −1 −2 |z| > 12z cos(ω ) + z 1 − z−1 cos(ω0) 0 z−1 sin(ω0) 0 1 − 2z−1 cos(ω ) + z−2 |z| > 1 1 − az−1 cos(ω0) a cos(ω0n)u[n] 0 an sin(ω0n)u[n] 1 − 2az−1 cos(ω ) + a2z−2 |z| > |a| az−1 sin(ω0) 1 − a2z−1 cos(ω ) + a2z−2 |z| > |a| Z x[n] ←→ X(z) Z and y[n] ←→ Y (z) Property Time domain Z-domain Linearity Ax[n] + By[n] AX(z) + BY (z) Time Shifting x[n − n0] X(z)z−n0 Z-scaling anx[n] X(a−1z) Conjugation x∗[n] X∗(z∗) Time Reversal x[−n] X(z−1) Convolution x[n] ∗ y[n] X(z)Y (z) dz Differentiation in z-domain nx[n] −zdX(z)

Solution. M1 β(1)[n] = δ[n] +3δ[n − 1]+ 5δ[n − 2, M2 ] β(2)[n] = 4, M3 4δ[n − 1] α(2)[n] = 4. F1(ω) = H(z) F2(ω) = G(z)H(z2) F3(ω) = G(z)G(z2) 2δ[n − 1] Discrete-Time Fourier Transform : X(ω) = n=Σ−∞ x[n]e−jωn ∞ ∫ Inverse Discrete-Time Fourier Transform : x[n] = 1 2π X(ω)ejωt dω . 2π x[n] X(ω) condition anu[n] 1 1 − ae−jω (n + 1)anu[n] 1 (1 − ae−jω )2 (n + r − 1)! anu[n] 1 |a| < 1 |a| < 1 |a| < 1 n!(r − 1)! (1 − ae−jω )r δ[n] 1 δ[n − n0] e−jωn0 ∞ x[n] = 1 2π k=Σ−∞ δ(ω − 2πk) ∞ 1 u[n] + 1 − e−jω ∞ k=Σ−∞ πδ(ω − 2πk) ejω0n 2π k=Σ−∞ ∞ δ(ω − ω0 − 2πk) cos(ω0n) π k=Σ−∞ ∞ ∞ {δ(ω − ω0 − 2πk) + δ(ω + ω0 − 2πk)} π j sin(ω n) Σ {δ(ω − ω k=−∞ − 2πk) − δ(ω + ω − 2πk)} ∞ k=Σ−∞ ∞ N k=−∞ 2π δ[n − kN ] N k=Σ−∞ δ ω .ω − 2πk Σ x[n] = ,,,  1 , |n| ≤ N ,  0 , |n| > N N sin(ω(N + 1/2)) sin(ω/2) sin(Wn) W Wn  . Σ X(ω) = 1 , 0 ≤ |ω| ≤ W , = sinc πn π π , 0 , W < |ω| ≤ π X(ω) is periodic with period 2π x[n] D←T→F T X(ω) and y[n] D←T→F T Y (ω) Property Time domain DTFT domain Linearity Ax[n] + By[n] AX(ω) + BY (ω) Time Shifting x[n − n0] X(ω)e−jωn0 Frequency Shifting x[n]ejω0n X(ω − ω0) Conjugation x∗[n] X∗(−ω) Time Reversal x[−n] X(−ω) ∫ Convolution x[n] ∗ y[n] X(ω)Y (ω) Multiplication x[n]y[n] 1 2π − X(θ)Y (ω θ)dθ 2π Differencing in Time x[n] − x[n − 1] (1 − e−jω)X(ω) 1− −jωe Accumulation Σk∞=−∞ x[k] 1 + πX(0) Σ∞k=−∞ δ(ω − 2πk) dX(ω) Frequency Differentiation nx[n] j dω 2π 2π Parseval’s Relation for Aperiodic Signals Σ∞k=−∞ |x[k]|2 1 ∫ |X(ω)|2dω Z-Transform : X(zsin(Wn) = n=Σ−∞ x[n]z−n ∞ I Inverse Z-Transform : x[n] = 1 2πj X(z)zn−1 dz . C x[n] X(ω) ROC anu[n] 1 1 − az−1 |z| > |a| n −a u[−n − 1] 1 − az−1 |z| < |a| nanu[n] n az−1 (1 − az−1)2 |z| > |a| az−1 −na u[−n − 1] (1 − az−1)2 |z| < |a| δ[n] 1 All z δ[n − n0] z−n0 All z u[n] cos(ω0n)u[n] sin(ω0n)u[n] n 1 − z−1 |z| > 1 1 − −1 −2 |z| > 12z cos(ω ) + z 1 − z−1 cos(ω0) 0 z−1 sin(ω0) 0 1 − 2z−1 cos(ω ) + z−2 |z| > 1 1 − az−1 cos(ω0) a cos(ω0n)u[n] 0 an sin(ω0n)u[n] 1 − 2az−1 cos(ω ) + a2z−2 |z| > |a| az−1 sin(ω0) 1 − a2z−1 cos(ω ) + a2z−2 |z| > |a| Z x[n] ←→ X(z) Z and y[n] ←→ Y (z) Property Time domain Z-domain Linearity Ax[n] + By[n] AX(z) + BY (z) Time Shifting x[n − n0] X(z)z−n0 Z-scaling anx[n] X(a−1z) Conjugation x∗[n] X∗(z∗) Time Reversal x[−n] X(z−1) Convolution x[n] ∗ y[n] X(z)Y (z) dz Differentiation in z-domain nx[n] −zdX(z)πn W Wn

Solution. M1 β(1)[n] = 2, M2 δ[n] + 3δ[n − 1] β(2)[n] = 4, M3 2δ[n − 1] α(2)[n] = 4. F1(ω) = H(z) F2(ω) = G(z)H(z2) F3(ω) = G(z)G(z2) 4δ[n − 1] Discrete-Time Fourier Transform : X(ω) = n=Σ−∞ x[n]e−jωn ∞ ∫ Inverse Discrete-Time Fourier Transform : x[n] = 1 2π X(ω)ejωt dω . 2π x[n] X(ω) condition anu[n] 1 1 − ae−jω (n + 1)anu[n] 1 (1 − ae−jω )2 (n + r − 1)! anu[n] 1 |a| < 1 |a| < 1 |a| < 1 n!(r − 1)! (1 − ae−jω )r δ[n] 1 δ[n − n0] e−jωn0 ∞ x[n] = 1 2π k=Σ−∞ δ(ω − 2πk) ∞ 1 u[n] + 1 − e−jω ∞ k=Σ−∞ πδ(ω − 2πk) ejω0n 2π k=Σ−∞ ∞ δ(ω − ω0 − 2πk) cos(ω0n) π k=Σ−∞ ∞ {δ(ω − ω0 − 2πk) + δ(ω + ω0 − 2πk)} π j sin(ω n) Σ {δ(ω − ω − 2πk) − δ(ω + ω − 2πk)} ∞ k=Σ−∞ ∞ N k=−∞ 2π δ[n − kN ] N k=Σ−∞ δ ω − 2πk x[n] = ,,, 1 , |n| ≤ N , 0 , |n| > N sin(ω(N + 1/2)) sin(ω/2) sin(Wn) W Wn X(ω) = 1 , 0 ≤ |ω| ≤ W , = sinc πn π π , 0 , W < |ω| ≤ π X(ω) is periodic with period 2π x[n] D←T→F T X(ω) and y[n] D←T→F T Y (ω) Property Time domain DTFT domain Linearity Ax[n] + By[n] AX(ω) + BY (ω) Time Shifting x[n − n0] X(ω)e−jωn0 Frequency Shifting x[n]ejω0n X(ω − ω0) Conjugation x∗[n] X∗(−ω) Time Reversal x[−n] X(−ω) ∫ Convolution x[n] ∗ y[n] X(ω)Y (ω) Multiplication x[n]y[n] 1 2π − X(θ)Y (ω θ)dθ 2π Differencing in Time x[n] − x[n − 1] (1 − e−jω)X(ω) 1− −jωe Accumulation Σk∞=−∞ x[k] 1 + πX(0) Σ∞k=−∞ δ(ω − 2πk) dX(ω) Frequency Differentiation nx[n] j dω 2π 2π Parseval’s Relation for Aperiodic Signals Σ∞k=−∞ |x[k]|2 1 ∫ |X(ω)|2dω Z-Transform : X(z) = n=Σ−∞ x[n]z−n ∞ I Inverse Z-Transform : x[n] = 1 2πj X(z)zn−1 dz . C x[n] X(ω) ROC anu[n] 1 1 − az−1 |z| > |a| n −a u[−n − 1] 1 − az−1 |z| < |a| nanu[n] n az−1 (1 − az−1)2 |z| > |a| az−1 −na u[−n − 1] (1 − az−1)2 |z| < |a| δ[n] 1 All z δ[n − n0] z−n0 All z u[n] cos(ω0n)u[n] sin(ω0n)u[n] n 1 − z−1 |z| > 1 1 − −1 −2 |z| > 12z cos(ω ) + z 1 − z−1 cos(ω0) 0 z−1 sin(ω0) 0 1 − 2z−1 cos(ω ) + z−2 |z| > 1 1 − az−1 cos(ω0) a cos(ω0n)u[n] 0 an sin(ω0n)u[n] 1 − 2az−1 cos(ω ) + a2z−2 |z| > |a| az−1 sin(ω0) 1 − a2z−1 cos(ω ) + a2z−2 |z| > |a| Z x[n] ←→ X(z) Z and y[n] ←→ Y (z) Property Time domain Z-domain Linearity Ax[n] + By[n] AX(z) + BY (z) Time Shifting x[n − n0] X(z)z−n0 Z-scaling anx[n] X(a−1z) Conjugation x∗[n] X∗(z∗) Time Reversal x[−n] X(z−1) Convolution x[n] ∗ y[n] X(z)Y (z) dz Differentiation in z-domain nx[n] −zdX(z)