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[Periodicity of DT signals, Aliasing] Suppose that we sample a CT signal

x(t) = 4 cos (18πt +π/4)
at a rate ω_s = 22π rad/s, to produce a DT signal x[n].
(a) Is the DT signal periodic? If so, what is its period in samples?
(b) The samples x[n] entered into a perfect D/A converter that always reconstructs waveforms assuming frequencies are assignable to the primary band. The output is
y(t) = A cos(ω₀t+θ₀) where ω₀ >0. Find the numbers A, ω₀, and θ₀.. Show all work.

User Lenkite
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1 Answer

3 votes

Main answer:

(a) The DT signal x[n] is periodic.

Step-by-step explanation:

To determine if the DT signal x[n] is periodic, we need to check if there exists a positive integer N such that x[n] = x[n+N] for all values of n.

Given that x(t) = 4 cos (18πt + π/4), we can find the period of the CT signal x(t) by dividing the angular frequency of the cosine function (18π) by the fundamental frequency (ω₀) of the signal. In this case, the fundamental frequency is the coefficient of t, which is 18π.

The period T of the CT signal is given by T = 2π/ω₀.

Now, let's determine the period of the DT signal x[n]. The period of a DT signal is equal to the reciprocal of the sampling rate, which is given as ω_s = 22π rad/s.

Therefore, the period of the DT signal x[n] is T_s = 2π/ω_s.

To find the period in samples, we need to divide the period T_s by the sampling interval Δt = 1, since the signal is sampled at a rate of 1 sample per unit time.

Therefore, the period in samples is N = T_s/Δt.

Substituting the values, we have N = (2π/ω_s) / 1 = 2π/ω_s.

Hence, the DT signal x[n] is periodic with a period of 2π/ω_s samples.

(b) To find the numbers A, ω₀, and θ₀, we can use the output y(t) = A cos(ω₀t + θ₀).

Given that the CT signal x(t) = 4 cos (18πt + π/4), we can see that y(t) is a reconstructed waveform of the DT signal x[n] through a perfect D/A converter.

Comparing the two equations, we can see that A = 4, ω₀ = 18π, and θ₀ = π/4.

Therefore, the numbers A, ω₀, and θ₀ are 4, 18π, and π/4, respectively.

I hope this explanation helps! Let me know if you have any further questions.

User Nzajt
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