Topic - Signal and system matlab

Project Description -

The first set of samples is from a signal x₁(t) consisting of broadband random noise and some periodic electromagnetic interference (EMI) x_e(t). The samples are taken at a rate of f_s1 samples/second and f_s1 is in your parameter file. The second set of samples is from a signal x₂(t) with broadband noise plus a narrowband signal x_s(t). The samples are taken at a rate of f_s2 samples/second and f_s2 is in your parameter file. The parameter file also has information about completing other parts of this project.

I. Sampling Effects

1. Sample the continuous-time signal x_ct(t) = Asin(2πf₀t) to form the discrete-time signal x_dt[n], at the sampling rates f_sa, f_sb and f_sc where A, f₀, f_sa, f_sb and f_sc are given in your parameter file. Include in your report three graphs of x_ct(t) vs. t and x_dt[n] vs. n for the three sampling rates. For each graph the discrete-time range should be 0 ≤ n ≤ 100 and the continuous-time range should be 0 ≤ t ≤ 100T_s.

2. The set of samples in xxxxxSig1.txt represents the discrete-time signal x₁[n], formed by sampling the continuous-time signal x₁(t). Find the best approximation to the CTFT X₁(f) of x₁(t ) using the DFT X₁[k] of x₁[n]. Treat x₁(t) as one period of a periodic signal. Then X₁(f) should consist entirely of impulses. There will be many small impulses caused by the random noise background and a smaller number of large impulses caused by the periodic EMI. Make the best estimate you can of the EMI x_e(t) by taking the inverse CTFT of the large impulses. In the report include these graphs:

a. The best estimate of x₁(t) over the time range in xxxxxSig1.txt.

b. The best estimate of |X₁(f)| over the frequency range 0 ≤ f ≤ f_s1 / 8 Hz.

c. The best estimate of ∠X₁(f) over the frequency range 0 ≤ f ≤ f_s1 / 8 Hz.

d. The best estimate of x_e(t) over the time range in xxxxxSig1.txt.

Also list the EMI frequencies and the values of X₁(f) at those frequencies.

3. Decimate the original x₁[n] by a factor of 4, forming a new x₁[n] which is a set of samples taken from x₁(t) at a sampling rate of f_s1/4 samples/second and repeat step 2. For the graphs of |X₁(f)| and ∠X₁(f), use a frequency range of 0 ≤ f ≤ f_s1/ 8 Hz.

4. Decimate the original x₁[n] by a factor of 16, forming a new x₁[n] which is a set of samples taken from x₁(t) at a sampling rate of f_s1/ 16 samples/second and repeat step 2. For the graphs of |X₁(f)| and ∠X₁(f), use a frequency range of 0 ≤ f ≤ f_s1/ 32 Hz.

For parts 2, 3 and 4, compare the frequencies of the EMI found in each case. If there are any differences, explain why.

II. Effects of Different Kinds of Filtering

1. Using the Laplace transform, find the time-domain response y_L(t) of a system with transfer function

H(s) = 2ζω_n/s²+ 2ζω_ns+ω_n² to the excitation x(t) = Acos(2πf₀t)u(t). The values of ζ, ω_n, A and f₀ are given in your parameter file.

2. Using the CTFT, find the time-domain response y_F(t) of a system with transfer function H(s) = 2ζω_n/s²+2ζω_ns+ω_n² to the signal x(t) = Acos(2πf₀t). The values of ζ, ω_n, A and f₀ are the same as in part 1.

Include a graph of y_L(t), y_F(t) and the excitation x(t). The graph should cover a time range of 0 ≤ t ≤10 / f₀.

3. The set of samples in xxxxxSig2.txt represents the discrete-time signal x₂[n], formed by sampling the continuous-time signal x₂(t). Find the best approximation to the CTFT X₂(f) of x₂(t) using the DFT X₂[k] of x₂[n]. Treat x₂(t) as one period of a periodic signal. Then X₂(f) should consist entirely of impulses. There will be many small impulses caused by the random noise background and a cluster of larger impulses caused by the narrowband signal. Make your best estimate of the lower and upper corner frequencies f_lo and f_hi of the narrowband signal.

4. Then simulate continuous-time filtering x₂(t) in two ways:

Method 1 - Frequency-domain Filtering

In this method simply multiply X₂(f) by the transfer function of an ideal band-pass filter between f_lo and f_hi to form X_2f(f). Then inverse transform the result back to the time domain as x_2f(t) . Use X₂[k] to approximate X₂(f) and use x₂[n] to approximate x₂(t).

Method 2 - Time-domain Filtering

Simulate time-domain filtering x₂(t) with an N = 4 Butterworth band-pass filter using the "butter" command and the "filter" command in MATLAB. Call the result of this filtering x_2t(t). In this method get the numerator and denominator coefficients of a digital filter that simulates the desired analog band-pass filter, filter x₂[n] with that digital filter and use the filtered x₂[n] as an approximation to the filtered x₂(t) . Use a command similar to this one to get the coefficients of the digital filter:

[b, a] = butter(4, [flo, fhi]/(fs/2)) ;

where "b" is the set of numerator coefficients, "a" is the set of denominator coefficients of the digital filter, "4" is the filter order, "flo" and "fhi" are f_lo and f_hi in Hz and "fs" is the sampling rate in samples/second. Use a command similar to this one to do the actual digital filtering:

x2t = filter(b,a,x2) ;

where "x2t" is the vector representing the result of the band-pass filtering, "b" is the set of numerator coefficients, "a" is the set of denominator coefficients of the digital filter and "x2" is a vector representing the original signal before filtering.

In the report include these graphs.

a. The best estimate of |X₂(f)|_dB vs f before filtering. For the graph of |X₂(f)|_dB use a frequency range of 0 ≤ f ≤ f_s2/ 8 Hz.

b. The best estimate of |X_2f(f)|_dB vs f after filtering. For the graph of |X_2f(f)|_dB use a frequency range of 0 ≤ f ≤ f_s2/ 8 Hz.

c. The best estimate of x_2f(t) over the time range in xxxxxSig2.txt.

d. The best estimate of x_2t(t) over the time range in xxxxxSig2.txt.

The two results x_2f(t) and x_2t(t) are very similar, but not identical. x_2f(t) looks a lot like a delayed version of x_2f(t). Explain why. Also, in the first few milliseconds and in the last few milliseconds, the two signals differ in another way. Explain why.

Textbook - SIGNALS AND SYSTEMS - Analysis Using Transform Methods and MATLAB, Second Edition by M. J. Roberts.

Please check chapter 9, 10 and 11 for code references that will help.

Attachment:- Assignment Files.rar