In the Goertzel algorithm for computation of the discrete Fourier transform, X[k] is com...

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 P9.49. 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 in both the real part and the imaginary part of X[k].