Lab - Discrete-Time Signals

Theory:

Signals are broadly classified into two categories, analog and discrete signals. An analog signal is continuous and has values at any point. A discrete signal is non- continuous and only has values at certain points. A discrete signal can be created from an analog signal by sampling at intervals. Typically, these intervals are equally spaced, but this is not a requirement.

Prelab Assignment:

Lab:

Generate the following sequences using the basic MATLAB signal functions and the basic M ATLAB signal operations discussed in this chapter. Plot signal samples using the stem function.

1. x₁(n) = 3δ(n+2) + 2δ(n) - δ(n-3) + 5δ(n-7), -5 ≤ n ≤ 15

2. x₄(n) = e^0.1n[u(n+20) - u(n-10)]

3. x₇(n) = e^-0.05n sin(0.1Πn + Π/3), 0 ≤ n ≤ 100.

Generate the following random sequences and obtain their histogram using the hist function with 100 bins. Use the bar function to plot each histogram.

1. x₁(n) is a random sequence whose samples are independent and uniformly distributed over [0, 2] interval. Generate 100,000 samples.

2. x₂(n) is a Gaussian random sequence whose samples are independent with mean 10 and variance 10. Generate 10,000 samples.

3. x₄(n) = _k=1Σ⁴ y_k(n) where each random sequence y_k(n) is independent of others with samples uniformly distributed over [-0.5, 0.5]. Comment on the shape of this histogram.

Generate the following periodic sequences and plot their samples (using the stem function) over the indicated number of periods.

1. x₁‾(n) = {...., -2, -1, 0, 1, 2,.....}_periodic. Plot 5 periods.

2. x₃‾(n) = sin(0.1Πn)[u(n) - u(n-10)] Plot 4 periods.

Let x(n) = {2, 4, -3, 1, -5, 4, 7}
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Generate and plot the samples (use the stem function) of the following sequences.

1. x₁(n) = 2x(n-3) + 3x (n+4) - x(n)

2. x₄(n) = 2e^0.5nx(n) + cos(0.1Πn)x(n+2), -10 ≤ n ≤ 10

The complex exponential sequence e^jω0n or the sinusoidal sequence cos(ω₀n) are periodic if the normalized frequency f₀ ^Δ₌ ω0/2Π is a rational number; that is, f₀ = K/N, where K and N are integers.

1. Prove the above the result.

2. Generate exp(j0.1Πn), -100 ≤ n ≤ 100. Plot its real and imaginary parts using the stem function. Is this sequence periodic? If it is, what is its fundamental period? From the examination of the plot, what interpretation can you give to the integers K and N above?

3. Generate and plot cos(0.1n), -20 ≤ n ≤ 20. Is this sequence periodic? What do you conclude from the plot? If necessary, examine the values of the sequence in MATLAB to arrive at your answer.