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2.2 A discrete-time signal x(n) is shown in Fig. P2.2. Sketch and label carefully each of the following signals. Figure P2.2 (a) x(n - 2) (b) x(4 - n) (c) x(n + 2) (d) x(n)u(2 - n) (e) x(n - 1)δ(n - 3) (f) x(n 2) (g) even part of x(n) (h) odd part of x(n)

2.2 A discrete-time signal x(n) is shown in Fig. P2.2. Sketch and label carefully each of the following signals. Figure P2.2 (a) x(n - 2) (b) x(4 - n) (c) x(n + 2) (d) x(n)u(2 - n) (e) x(n - 1)δ(n - 3) (f) x(n 2) (g) even part of x(n) (h) odd part of x(n)

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2.2 A discrete-time signal x(n) is shown in Fig. P2.2. Sketch and label carefully each of the following signals. Figure P2.2 (a) x(n - 2) (b) x(4 - n) (c) x(n + 2) (d) x(n)u(2 - n) (e) x(n - 1)δ(n - 3) (f) x(n 2) (g) even part of x(n) (h) odd part of x(n)

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CS Amplifiers vs. CG Amplifiers [19 points] (a) (b) Fig. P2 All NMOS transistors are biased at saturation region, but only small-signal equivalent circuits are shown in the figures. Each individual NMOS transistor has C2d = 0.1•Cgs, Cdb = Csb = 0.2•Cgs and intrinsic gain gmr0 = 40(>>1). However, the transistor is used as a CS amplifier in Fig. P2(a), but a CG amplifier in Fig. P2(b). Note that the intrinsic output impedance, r8, of the NMOS transistor still exists even though the value of Rsig and RL. have been assigned as rd/2 and re, respectively. For the following questions, please include device capacitances Cb, C2 d, Cb, and Cb if necessary. (a) Find the low frequency gain |vo/vi| of the CS amplifier in Fig. P2(a). (b) Use OCTC method only to find the expression of the time-constant, τ1,cs in terms of Cpd and ro, across the gate and source of the NMOS in Fig. P2(a). (c) Use OCTC method only to find the expression of the time-constant, τ22Cs in terms of Cpd and ro, across the drain and source of the NMOS in Fig. P2(a). (d) Use OCTC method only to find the expression of the time-constant, τ3, cs in terms of C84 and ro, across the drain and gate of the NMOS in Fig. P2(a). (e) Based on (b), (c), and (d), find the expression of 3-dB frequency, fucs in terms of Cp4 and rC, of the CS amplifier. (f) Find the low-frequency gain |v9/vi | of the CG amplifier in Fig. P2(b). (g) Use OCTC method only to find the expression of the time-constant, τ1,CG in terms of Cpd and r6, across the source and gate of the NMOS in Fig. P2(b). (h) Use OCTC method only to find the expression of the time-constant, τ2CG in terms of C84 and ro, across the drain and gate of the NMOS in Fig. P2(b). (i) Based on (g) and (h), find the expression of 3-dB frequency, f fh,co in terms of CBd and r0, of the CG amplifier. (j) Based on (a) and (f), find the ratio of low-frequency gains, |Av,co/Av, cs|. (k) Based on (c) and (i), find the ratio of 3-dB frequencies, fH,co/fH,cs. Result Summary: (a) Av,Cs = (b) τ1,Cs = (c) τ2,Cs = (d) τ3,Cs = (e) fH,CS = (f) Av,CG = (g) τ1,CG = (h) τ2,CG = (i) fH,CG = (j) |Av,CG/Av,CS| = (k) fH,CG/fH,CS =CS Amplifiers vs. CG Amplifiers [19 points] (a) (b) Fig. P2 All NMOS transistors are biased at saturation region, but only small-signal equivalent circuits are shown in the figures. Each individual NMOS transistor has C2d = 0.1•Cgs, Cdb = Csb = 0.2•Cgs and intrinsic gain gmr0 = 40(>>1). However, the transistor is used as a CS amplifier in Fig. P2(a), but a CG amplifier in Fig. P2(b). Note that the intrinsic output impedance, r8, of the NMOS transistor still exists even though the value of Rsig and RL. have been assigned as rd/2 and re, respectively. For the following questions, please include device capacitances Cb, C2 d, Cb, and Cb if necessary. (a) Find the low frequency gain |vo/vi| of the CS amplifier in Fig. P2(a). (b) Use OCTC method only to find the expression of the time-constant, τ1,cs in terms of Cpd and ro, across the gate and source of the NMOS in Fig. P2(a). (c) Use OCTC method only to find the expression of the time-constant, τ22Cs in terms of Cpd and ro, across the drain and source of the NMOS in Fig. P2(a). (d) Use OCTC method only to find the expression of the time-constant, τ3, cs in terms of C84 and ro, across the drain and gate of the NMOS in Fig. P2(a). (e) Based on (b), (c), and (d), find the expression of 3-dB frequency, fucs in terms of Cp4 and rC, of the CS amplifier. (f) Find the low-frequency gain |v9/vi | of the CG amplifier in Fig. P2(b). (g) Use OCTC method only to find the expression of the time-constant, τ1,CG in terms of Cpd and r6, across the source and gate of the NMOS in Fig. P2(b). (h) Use OCTC method only to find the expression of the time-constant, τ2CG in terms of C84 and ro, across the drain and gate of the NMOS in Fig. P2(b). (i) Based on (g) and (h), find the expression of 3-dB frequency, f fh,co in terms of CBd and r0, of the CG amplifier. (j) Based on (a) and (f), find the ratio of low-frequency gains, |Av,co/Av, cs|. (k) Based on (c) and (i), find the ratio of 3-dB frequencies, fH,co/fH,cs. Result Summary: (a) Av,Cs = (b) τ1,Cs = (c) τ2,Cs = (d) τ3,Cs = (e) fH,CS = (f) Av,CG = (g) τ1,CG = (h) τ2,CG = (i) fH,CG = (j) |Av,CG/Av,CS| = (k) fH,CG/fH,CS =
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