Lab - ADC and DAC

Theory:

Analog to digital and digital to analog conversion are extremely important in DSP applications. The representation of an analog signal in computer memory requires analog to digital conversion, requiring sampling and quantization. The playback or production of an analog signal from computer memory is known as digital to analog conversion. This requires reconstruction of a signal from sampled points and involves some form of interpolation. Depending on how the digital signal is stored, the analog signal can either be reconstructed completely (lossless), or some information must be created to fill in the gaps (lossy).

Prelab Assignment:

Lab:

1. An analog signal x_a(t) = sin(1000Πt) is sampled using the following sampling intervals. In each case, plot the spectrum of the resulting discrete-time signal.
a. T_s = 0.1 msec
b. T_s = 1 msec
c. T_s = 0.01 sec

2. Consider an analog signal x_a(t) = cos(20Πt), 0 ≤ t ≤ 1. It is sampled at T_s = 0.01, 0.05, and 0.1 sec intervals to obtain x(n).
a. For each T_s, plot x(n).
b. Reconstruct the analog signal y_a(t) from the samples x(n) using the sinc interpolation (use ?t = 0.001) and determine the frequency in ya(t) from your plot. (Ignore the end effects.)
c. Reconstruct the analog signal y_a(t) from the samples x(n) using the cubic spline interpolation, and determine the frequency in y_a(t) from your plot. (Again, ignore the end effects.)
d. Comment on your results.

3. Consider an analog signal x_a(t) = cos(20Πt + θ), 0 ≤ t ≤ 1. It is sampled at Ts = 0.5sec intervals to obtain x(n). Let θ = 0, Π/6, Π/4, Π/3, Π/2. For each of these θ values, perform the following.

a. Plot x_a(t) and superimpose x(n) on it using the plot(n,x,'o') function.
b. Reconstruct the analog signal ya(t) from the samples x(n) using the sinc interpolation (Use ?t = 0.001) and superimpose x(n) on it.
c. Reconstruct the analog signal ya(t) from the samples x(n) using the cubic spline interpolation and superimpose x(n) on it.
d. You should observe that the resultant reconstruction in each case has the correct frequency but a different amplitude. Explain this observation. Comment on the role of phase of xa(t) on the sampling and reconstruction of signals.
4. Consider a signal with spectrum

X(ω) = { non-zero, |ω| ≤ ω₀;

          {  0,           ω₀ < |ω| ≤ Π

a. Show that the signal x(n) can be recovered from its samples x(mD) if the sampling frequency ω_s = 2Π/D ≥ 2ω₀.
b. Sketch the spectra of x(n) and x(mD) for D = 4.
c. Show that x(n) can be reconstructed from the bandlimited interpolation

_k=-∞∑^∞x(kD) sinc[f_c(n - kD)];  f_c = 1/D

For ω₀ < 2Πf_c < ω_s - ω₀

5. Using the function interp, study the operation of factor-of-4 interpolation on the following sequences. Use the stem function to plot the original and the interpolated sequences. Experiment, using the filter length parameter values equal to 3 and 5. Comment on any differences in performance of the interpolation.
a. x₁(n) = sin(0.6Πn), 0 ≤ n ≤ 100.
b. x₂(n) = sin(0.8Πn) + cos(0.5Πn), 0 ≤ n ≤ 100.
c. x₃(n) = 1 + cos(Πn), 0 ≤ n ≤ 100.
d. x₄(n) = 0.2n, 0 ≤ n ≤ 100.
e. x₅(n) = {1, 1, 1, 1, 0, 0, 0, 0, 0, 0}_PERIODIC, 0 ≤ n ≤ 100.