Consider a real, bandlimited signal xa(t) whose Fourier transform Xa(jΩ) has the followi...

Consider a real, bandlimited signal x_a(t) whose Fourier transform X_a(jΩ) has the following property:

X _a (jΩ) = 0 for | Ω| > 2π · 10000.

That is, the signal is bandlimited to 10 kHz.

We wish to process x_a(t) with a highpass analog filter whose magnitude response satisfies the following specifications (see Figure P7.33):

where Ω_s and Ω_p denote the stopband and passband frequencies, respectively.

(a) Suppose the analog filter Ha(jΩ) is implemented by discrete-time processing, according to the diagram shown in Figure 7.2.

The sampling frequency f_s = 1/T is 24 kHz for both the ideal C/D and D/C converters. Determine the appropriate filter specification for |H(e^jω)|, the magnitude response of the digital filter.

(b) Using the bilinear transformation , we want to design a digital filter whose magnitude response specifications were found in part (a). Find the specifications of |GHP (j Ω₁A|, the magnitude response of the highpass analog filter that is related to the digital filter through the bilinear transformation. Again, provide a fully labelled sketch of the magnitude response specifications on |GHP (jΩ₁)|.

(c) Using the frequency transformation (i.e., replacing the Laplace transform variable s by its reciprocal), design the highpass analog filter G_HP (jΩ₁) from the lowest-order Butterworth filter, whose magnitude-squared frequency response is given below:

In particular, find the lowest filter order N and its corresponding cutoff frequency Ω c, such that the original filter’s passband specification (|Ha(jΩ_p)| = 0.9) is met exactly. In a diagram, label the salient features of the Butterworth filter magnitude response that you have designed.

(d) Draw the pole–zero diagram of the (lowpass) Butterworth filter G(s₂), and find an expression for its transfer function.