A filter is needed to recover x2(n) from the signal, x(n) = x1(n) + x2(n) + x3(n

Question

A filter is needed to recover x2(n) from the signal, x(n) = x1(n) + x2(n) + x3(n), where x1(n) = sin(.5n), x2(n) = sin(1.4n), and x3(n) = cos(2.7n). (a) What kind of filter is required, LP,BP,HP, or BR ? (b) Specify a filter cut-off frequency or frequencies in radians, and find the corresponding cut-off sample or samples, k1 etc, for H(k). (c) Assume that H(k) is ideal in the sense that each of its samples is either 1 or 0. Find a closed form for the impulse response h(n) using H(k), the inverse DFT operation, and the cut-off sample symbol(s) ki. (d) Find a real expression for h(n) from part (c).

Explanation / Answer

(a) w1 = frequency of x1= 0.5 rad/ sample

  w2 = frequency of x2= 1.4 rad/ sample

w3= frequency of x3= 2.7 rad/ sample

To filter the 1.4 rad/s frequency, BP filter is used

(b) Wc1, Wc2= cutoff frequency

0.5<Wc1<1.4 , 1.4<Wc2<2.7,

Let Wc1=1.3, and Wc2=1.5

(C) H(e^jw)= 1 , 1.3<w<1.5

=0, else

h(n)= [sin(wc2)-sin(wc1)]/(pi*n)

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