Question

Consider the cascade of LTI discrete-time systems shown in Figure P2.37. LTI System 1 hi[n], H (el) LTI System 2 h2[n], H2(eje) Figure P2.37 The first system is described by the frequency response Hi(j =c-joo < 0.25% 11 0.25% < and the second system is described by <A hain) = 2 Sin(0.57) (a) Determine an equation that defines the frequency response, H(e)®), of the overall system over the range -- SUSA. (b) Sketch the magnitude. He"), and the phase, ZH(e)), of the overall frequency response over the range - USA. (c) Use any convenient means to determine the impulse response h[n] of the overall cas- cade system.

Answer 1

Anscoer: (a) The frequency response of the system is, Hi (Cuco) ze jw so IWIS 0.25T I 0.25TT<IWIST The period of the frequency response of mod of the frequency response of the first system's 2a Observe the frequency response of the First system ar the high pass filter Apply inverse Fourier transform and obtain the imple respone of the first system. | btn] Sn-] - Sin 0,25 m (0-) Tn-il The plot of the magnitude response of the first system Hilejw) I 50 -0.12510 0.1251 621) 0.5 0.78T1 TT 0.250 0.1250 -0.50 -T -0.75 is (fig. 1) => The impulse response of the sccand system h2 [n] = ? Sin (osmn) The fourer transform of the sinc Tin function is rectangular signal Singen <> Si, Icousca Sin won OlcWc In O, Cocort

(2) frequency response of the Obtain the second system Hz (ejw) - 52 ICO) SOSTI 20 0.5 T <IWICIT ne plot of magnitude response of the second system + | Haleiw) 1 0.27 OST T -0.5 0.12511 0.25TI OSTI T -0.2571_0.1271 (fig. 2) => in Convolution in time domain is equal to multiplication Frequency domain H (ejo) = Hi (eo) Hz (630) solw.o.857 ] [S2 (wisoist , - {18.255 Kit LoosiciWIST twico.5T system ? ze sw, 0.257<1W120.57 0.5^<lwist the frequency response of the overall cascade represented by Heſto) is, Co wis0.251 H(esw) = 12e to 0.251 < 1W1 COST Lo, Ost <1WICH Coņ

(3) the magnitude response of o Obtain the overall system Heim) H[evw) 1 = Colwi 50255 120.251 < W<0.511 Lo 0.50 <1W1ST -> Plot of magnitude response of the overall cascade -0. 5T -0.25T O.250 OST | | (fig. 3) Obtain the phase response of the overall cascade system Co (W120.251 (eco) = (-0, 0.25T < 1W1So.si lo 0-5T < 1WIST The plot of the phase response of the overall cascoce system LH (JW) Soma- 0.571 0.250 0.251 OST -T - 0.5T 0,257 0 ł w -0.2571 -0.511 Sranned with I fig.uy

> Observe figure 3, it represents the magnitude of the overall system. system considering the range of the prevall el cascade from 1 to TT, it represents a band pass filter consider a low pass filter ranging from -0.12571 to 0,1251 Hi, -0.1257 o 6.125T - (Fig.s) Perform frequency shifting of the signal shown in fig 5. colth a factor of wo = 40.37511 to obtain the magnitude response shown in figa of the system Hi (euw) is. The impulse response hl(n) = 2 Sino-125TH) Tin the Represent the response shown in fig 3. in terms of Frequency response of Hlexos. TH (6)] = Hz (e3(W-Q.375T)) +Hz (@j[640.375T1)

uency response of Include the phase term to get the frequency respon the overall system. Heim) = 1H (euw) leich Cejro) H Ceico - esto [itz Celfo-6. 3750))+ + Ces (+ 0-3750)7-> Consider the following founier transform theorem evwon xen] Fly X (ej (w-woo) Apply inverse founder transform to equation (1) the impulse Use the founier transform theorem and obtain response of the overall cascccle system. F" [4 Ccvas) - F" [e-it® [H2 (2160.0.315) iſC7 Como 597) ]] h{n} = he[n-11e +30.375T(n-1) 492 6-13 e 10.375 77(n-0 = hz (n-1)(et j0-3357(n-1) +e-3 0.375T (n-1)) = 2h2 (n-1) ſetj 0,3755(0-1) + 6-10.385 n(n-1) 7 2 -> Substitute herns and obtain the impuls response of the overall cascade systems. brorso fo Sin (0.125T(n-1)) 7 re+30.37571 (n-1) 48-36.375 (n-07 T(n-1) se a 2 sin (0.12511 (n-1) cos C0-375 11(n-1)] | 1 n-1) Thus, the impuls response of the overall costcode system 4 sin (0.12557 (0-1)) 1-17) cos (0.375 15(n-1)) T(n-1) Scanned with

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