Find the approximate fourier transform for the waveform

Question

Lab- Matlab - Fourier

Question 1

Find the Fourier transforms for the following waveforms. Plot the waveforms and their magnitude spectra. [Hint: Use Eq. (2-184)
(a) Π(t-3)/4.

(b) 2.

(c) ∧(t - 5)/5

Question 2

By using Eq, (2-184), find the approximate Fourier transform for the following waveform:

          sin(2Πt/512) + sin(70Πt/512), 5 < t < 75     

x(t) =
           0,                     t elsewhere

Question 3

Using a computer program, calculate the DFT of a rectangular pulse, Π(t). Take five samples of the pulse and pad it with 59 zeros so that a 64-point FFT algorithm can be used. Sketch the re-sulting magnitude speculum Compare this result with the actual spectrum for the pulse. Try other combinations of the number of pulse samples and zero-pads to see how the resulting FFT changes.

Lab - Matlab - SNR

Question 1

For a 4-bit PCM system, calculate and sketch a plot of the output SNR (in decibels) as a function of the relative input level, 20 log (x_rms/V) for
(a) A PCM system that uses μ = 10 law companding.
(b) A PCM system that uses uniform quantization (no companding).
(c) Which of these. systems is better to use in practice? Why?

Question 2
The performance of a μ = 255 law companded PCM system is to be examined when the input consists of a sine. wave having a peak value of V volts. Assume that M = 256.
(a) Find an expression that describes the output SNR for this companded PCM system.
(b) Plot (S/N)_out (in decibels) as a function of the relative input level, 20 log (x_rms,/V), Compare this result with that shown in Fig. 3-10.

Lab - Matlab - PSD

Question 1

Evaluate the ND for the waveform shown in Fig. P2-2.

Question 2
In Fig. 3-16, the PSDs for several line codes are shown. These PSDs were derived assuming unity power for each signal so that the PSDs could be compared on an equal transmission power basis. Rederive the PSDs for these line codes, assuming that the peak level is unity (i.e., A = 1). Plot the PSDs so that the spectra can be compared on an equal peak-signal-level basis.

Lab- Matlab - DC and RMS

Question 1

Use v(t) = ae^-at, a > 0, to approximate δ(t) as a → ∞
(a) Plot v(t) for a = 0_1, 1, and 10,
(hi Plot 1/(1) for a = 1, and 10,

Lab - Matlab- Bandpass signaling

Question 1

Let the transfer function of an ideal bandpass filter be given by

where B_T is the absolute bandwidth of the filter.

(a) Sketch the magnitude transfer function |H(f)|
(b ) Find an expression for the waveform at the output, v₂(t), if the input consists of the pulsed carrier

v₁(t) = AΠ(t/T) cos(ω_ct)

(c) Sketch the output waveform v₂(t) for the case when B_T= 4/T and f_c >> B_T.

(Hint: Use the complex-envelope technique, and express the answer as a function of the sine integral, defined by

 Si(u) = ₀∫^usinλ/λ.dλ

The sketch can be obtained by looking up values for the sine integral from published tables [Abramowitz and Stegun, 1964] or by numerically evaluating Si (u).

Question 2:

A bandpass filter as shown in fig P4-11 has the transfer function

H(s) = Ks/(s² + (ω₀/Q)s + ω₀²)

where Q = R√(C/L) the resonant frequency is f_o = 1/(2Π√LC), ω₀ = 2Πf₀, K is a constant, and values for R, L, and C are given in the figure. Assume that a bandpass signal with f_c = 4 kHz and a bandwidth of 200 Hz passes through the filter, where f₀ =f_c.

(a) Using Eq. (4-39), find the bandwidth of the filter.
(b) Plot the carrier delay as a function of f about f_o.
(c) Plot the group delay as a function of f about f_o.
(d) Explain why the filter does or does not distort the signal.

Lab - Matlab- Modulated signals and filters

Question 1

Plot the amplitude response, the phase response, and the phase delay as a function of frequency for the following low-pass filters, where B= 100 Hz:

(a) Butterworth filter, second order:

H(f) = 1/(1+√2(if/B) + (jf/B))²

(b) Butterworth filter, fourth order:

H(f) = 1/[(1+ 0.765(jf/B) + (jf/B)²][1 + 1.848(jf/B) + (jf/B)²]

Compare your results for the two filters.

Question 2

The nonlinear output-input characteristic of an amplifier is

v_out(t) = 5v_in(t) + 1.5v²_in(t) + 1.5v³_in(t)

Assume that the input signal consists of seven components:

V_in(t) = 1/2 + 4/Π² ∑⁶_k=1 1/(2k -1)².cos[(2k - 1)Πt]

(a) Plot the output signal and compare it with the linear output component 5v_in(t).

(b) Take the FFT of the output v_out(t), and compare it with the spectrum for the linear output component.