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113_1_block_diagram_examples

# 113_1_block_diagram_examples - 2.7(a(b From the above...

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Unformatted text preview: 2.7 (a) (b) From the above ﬁgure we get w[n] = h[0](x[n] + Bllx[n — l] + [321x[n — 2]) and y[n] = w[n] + [312w[n — 1] + [32214])”; — 2]. Making use of the ﬁrst equation in the second we arrive at y[n] = h[0](X[n] + 51 116D? , 1] + B21x[n , 2]) +[312h[0](X[n—1]+l311x[ﬂ — 21+ 52MB! —3]) + B22h[0](x[n — 2] + Bllxm — 3] —— [321x[n — 4]) = h[0](x[n]+([311+ [312)x[n —1]+(621+B12[311“ [322)x[n— 2] +(l312l321+ [32231 1 Mn — 3] + [3223211411 — 4])- (c) Figure P2.l(c) is a cascade of a ﬁrst-order section and a second-order section. The input-output relation remains unchanged if the ordering of the two sections is interchanged as shown below. The second-order section can be redrawn as shown below without changing its input- output relation. The second-order section can be seen to be cascade of two sections. Interchanging their ordering we ﬁnally arrive at the structure shown below: Analyzing the above structure we arrive at s[n] = O.6x[n] + 0.3x[n — 1] + 0.2x[n — 2], u[n] = s[n] — 0.8u[n — '1] — 0.5u[n — 2], y[n +1] : u[n] + 0.4y[n]. From u[n] : y[n +1] — 0.4y[n]. Substituting this in the second equation we get after some algebra y[n +1]: S[n] * 0.4y[n] * O.18y[n * 1] + 0.8y[n * 2]. Making use of the ﬁrst equation in this equation we ﬁnally arrive at the desired input-output relation y[n] + 0.4y[n— 'l] + 0.'18y[n — 2] — 0.2y[n — 3] = 0.6x[n — 1] + 0.3x[n — 2] + O.2x[n— 3]. ...
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113_1_block_diagram_examples - 2.7(a(b From the above...

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