Consider a five tap FIR filter with coefficients h(0) = h(4) = , h(1) = h(3) = a

Question

Consider a five tap FIR filter with coefficients h(0) = h(4) = , h(1) = h(3) = and h(2) = . The input to this filter is x(n) = [cos(0.2n) + cos(0.4n) + cos(0.7n)] u(n) where u(n) is the unit-step function. The filter must be designed such that the steady-state output is a cosine signal of frequency 0.4 radians/sample. The frequencies of 0.2 radians/sample and 0.7 radians/sample should be blocked.

(a) Find the magnitude and phase response of the filter as a function of , and .

(b) Is the phase linear? What is the group delay in samples?

(c) Write the difference equation relating the output y(n) to a general x(n) (not necessarily the input x(n) as given above). The difference equation will be in terms of , and . (

d) Solve for , and . Show the mathematical development. The actual solution can be done either on a calculator or using MATLAB.

Explanation / Answer

given

h(0)=h(4)=

h(1) = h(3) =

h(2) =

x(n) = [cos(0.2n) + cos(0.4n) + cos(0.7n)] u(n)

steady-state output is a cosine signal of frequency 0.4 radians/sample

frequencies to be blocked =0.2 radians/sample and 0.7 radians/sample

we know that

y[n] = _ m= h[m]x[n m]

y[n] = m= h[m]x[n m]

= m= h[m] cos! 0(n m) + 0 "

= ! m= h[m] cos(0m) " cos(0n + 0)

H(0) = n= h[n]

H() = H() = n= (1)nh[n]

H() = H2()H1() = H1()H2()

h[n] = (h1 h2)[n] H() = H1()H2() .

hL[n] = 1 L ! [n] + [n 1] + ··· + [n (L 1)]"

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