Question: In the Goertzel algorithm for computation of the discrete Fourier transform, X[k] is computed as X[k] = Yk[n], where yk[n] is the output of the

In the Goertzel algorithm for computation of the discrete Fourier transform, X[k] is computed as X[k] = Yk[n], where yk[n] is the output of the network shown in Figure. Consider the implementation of the Goertzel algorithm using fixed-point arithmetic with rounding. Assume that the register length is B bits plus the sign, and assume that the products are rounded before additions. Also, assume that round-off noise sources are independent.

(a) Assuming that x[n] is real, draw a flow graph of the linear ?noise model for the finite-precision computation of the real and imaginary parts of x[k]. Assume that multiplication by ? 1 produces no round-off noise.

(b) Compute the variance of the round-off noise t=in both the real part and the imaginary part of X[n].

x[r] valr) 2cos

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