Question 1:

A discrete cosine signal x[n] = cos[Π/8 n] is a periodic signal. Determine the period of this signal.

Question 2:

A discrete signal is given by x[n] = {1, 2, 3, 4}, i.e., x[0] = 1, x[1] = 2 and x[2] = 3 and x[3] = 4. Express the following signals as a linear combination of the unit sample sequence (e.g., x[n] = ∑x[m]δ[n - m]) and plot the signals.

(a) x[n]

(b) x₁[n] = x[n - 2].

Question 3:

The input-output relationship of a running average filter is given by the following difference equation.

y[n] = 1/4(x[n] + x[n - 1]+ x[n - 2] + x[n - 3] + x[n - 4])

(a) Determine the impulse response of the filter.

(b) Determine the output for an input signal

x[n] = {1,        0 ≤ n ≤ 10

         {0,        otherwise

Question 4:

A filter is given by its impulse response

h[n] = {1, 0 , -1}

where h[0] = 1.

(a) Determine the difference equation.

(b) Use convolution to calculate the output signal for an input signal x[n] = {1, 2, 3, 4, 4, 3, 2, 1}.

Question 5:

Determine whether the following system is a causal system

y[n] = x[n + 1] - x[n - 1]

Question 6:

An analog signal has a bandwidth of [0, 4000] Hz. It is processed by an analog low pass filter with a cutoff frequency of 1000Hz. The same signal processing task is to be performed by using DSP. Determine the minimum sampling frequency and the corresponding cutoff frequency of the digital low pass filter.

Question 7:

An analog signal is defined as the following

x(t) = 1 + 2 cos(100Πt) + 3 cos(2000Πt)

Determine the 3 frequency components: f₁, f₂ and f₃. Suppose the sampling frequency is f_s, = 4000Hz, determine the corresponding frequencies ω₁, ω₂ and ω₃ of the discrete signal.