Question

Given signal x(t) = sinc(t):

1. Find out the Fourier transform of x(t), find X(f), sketch them.

2. Find out the Nyquist sampling frequency of x(t).

3. Given sampling rate fs, write down the expression of the Fourier transform of xs(t), Xs(f) in terms of X(f).

4. Let sampling frequency fs = 1Hz.

Sketch the sampled signal xs(t) = x(kTs) and the Fourier transform of xs(t), Xs(f).

5. Let sampling frequency fs = 2Hz. Repeat 4.

6. Let sampling frequency fs = 0:5Hz. Repeat 4.

7. Let sampling frequency fs = 1:5Hz. Repeat 4.

8. Let sampling frequency fs = 2/3Hz. Repeat 4.

8.1 Design Matlab programs to illustrate items 4-8

8.2 plot all the graphs.

9. Design a Matlab function to calculate the Fourier transform
of a sampled signal

xs(t), Xs(f) = Σ k x(kTs) exp(-j2πf . kTs).

NOTE: In Matlab and this experiment, sinc(t) is defined as
sinc(t) = sin(π t)/(πt). Under

this definition: sinc(2Wt) 1/(2W) rect( f/2W )

Answer #1

Consider a signal x(t) = 20 sinc(10t) is to be represented by
samples using ideal (impulses) sampling.
a) Sketch the signal spectral density X(f).
b) Find the minimum sampling frequency that allows reconstruction
of x(t) from its samples.
c) Sketch the sampled spectrum Xs(f) in the range [-55, 55] with
sampling frequency fs = 25 Hz.

Consider speech signal x(t), which has the magnitude spectrum
X(f)= rect(f/100).
Sketch the magnitude spectrum of the sampled x(t) resulted by
conducting natural sampling with the sampling freq 180Hz. Comment
on your result.
Discuss how to perform digital to analog conversion to recover
x(t). Is it possible to fully recover x(t) without distortion?
Justify your answer.

The signal x(t)=sinc2(100t) is sampled with frequency
fs=150sample/sec 1. Determine the spectral representation of the
sampled signal (n=-3 to n=3). 2. Plot the spectral representation
of the sampled signal and determine if the signal has aliasing. 3.
Determine the minimum frequency of sampling (Nyquist sampling
frequency) so that the signal can be reconstructed using an ideal
low pass filter. Justify your answer.

Consider the signal x(t) = 3 cos 2π(30)t + 4 .
(a) Plot the spectrum of the signal x(t). Show the spectrum as a
function of f in Hz. ?π? For the remainder of this problem, assume
that the signal x(t) is sampled to produce the discrete-time signal
x[n] at a rate of fs = 50 Hz.
b Sketch the spectrum for the sampled signal x[n]. The spectrum
should be a function of the normalized frequency variable ωˆ over
the...

Find the forward discrete fourier transform of
f(t)={0,2,4,-2}

SUB-DSP
given the following function and sample rate:
1) x(t)=sin(200pi't')+2sin(900pi't') and Fs=600HZ
find
a)the nyquist rate
b)sampled signal,x(n) (taking into account aliasing if
present)
c)Reconstructed signal,y(t)

A signal described by y(t) = .25*sin(6000*t) + cos(3500*t) was
sampled with fs = 1 kHz. Determined:
a) Whether this signal was aliased. Provide the spectral plot
(magnitude versus frequency)
b) If the signal proved to be aliased, find the right sampling
rate
c) Provide the spectral plot indicating the right frequencies of
the signal

Find the Fourier Transform of the following, Show all
steps:
1- f(x)=e^(-6x^2)
2- f(x) is 0 for all x except 0≤x≤2 where f(x)=4

Question : Design the low and high pass filter for the signal,
x(t) = 10 sin (10 t) + 1 sin (1000 t) by MATLAB
Is below answer right?
at ?High pass , 5row
shouldn't this change from sin(100*t) ?
sin(1000*t)
x = 10*sin(10*t) + 1*sin(100*t); ? x = 10*sin(10*t) +
1*sin(1000*t); ???
.....................................................................................................................................................
?Low pass
clc;
rng default
Fs=2000;
t=linspace(0,1,Fs);
x=10*sin(10*t)+sin(1000*t)%given signal
n=0.5*randn(size(t));%noise
x1=x+n;
fc=150;
Wn=(2/Fs)*fc;
b=fir1(20,Wn,'low',kaiser(21,3));
%fvtool(b,1,’Fs’,Fs)
y=filter(b,1,x1);
plot(t,x1,t,y)
xlim([0 0.1])
xlabel('Time (s) ')
ylabel('Amplitude')
legend('Original Signal','Filtered Data')...

A waveform, x(t)=10cos(100 πt+π/3)+20cos(200 πt+π/6) is to be
uniformly sampled for a digital transmission.
(a) What is the maximum allowed time interval between sample values
to ensure perfect signal reproduction?
(b) Draw the discrete time signal obtained and frequency
transform after sampling.
(c) If we want to reproduce one hour of this waveform, how many
sample values need to be stored?

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