代写Digital Signal Processing and Digital Filters Practice Sheet 1代写C/C++语言-代写Algorithm 算法作业

Digital Signal Processing and Digital Filters

Practice Sheet 1

1) Which of the following statements is true and why? (Multiple choices may be correct)

i) All discrete-time signals are digital signals.

ii) All digital signals are discrete-time signals.

iii) Some discrete-time signals are digital signals.

iv) Some digital signals are discrete-time signals.

2) Let us define a sequence by,

x[n] = (r1)nu [n] − (r2 )nu [− (n + 1)] (1)

where r1 = −1/3 and r2 = 1/2.

• On the z-plane, plot the poles and zeros together with the region-of-convergence.

• In the above sequence in (1), what happens if we exchange r1 and r2?

3) For some linear-time-invariant system, the transfer function is given by,

Suppose x [n] is the input to the system and y [n] is the output.

Derive the difference equation that is satisfied by x [n] and y [n].

4) Filter Specification via Z-transform Suppose that a function is given by,

• Is ϕ (t) a symmetric function? Argue by plotting this function.

• Convert ϕ (t) into an FIR filter by sampling it at integer points, that is, ϕ (t) , t = k where k = 0, ±1, ±2, . . ..

Let Φ (z) be the z-transform of this FIR filter sequence. Write the explicit form of Φ (z).

• Inverse filter design. Suppose that the FIR filter is defined by

p [k] = ϕ (t)|t=k, k = Z (that is, k takes integer values).

Then, we say that pinv [k] is an inverse-filter when,

Identify the transfer function of pinv [k] in terms of Φ (z).

Write down the impulse-response of pinv [k] given the definition of ϕ (t). Is pinv [k] an FIR or IIR filter?

Plot the impulse response of pinv [k].

5) Filter Identification

The pole-zero plot of a discrete filter is given below.

When the input x [n] = 1 for all n, the output is exactly the same.

What is the impulse response of such a filter?