Electrical Communications Systems

Course No. ECE.09.433

Laboratory Project 1

Waveform Synthesis and Spectral Analysis

Objectives

This project has 4 parts. In Part 1, you will generate arbitrary waveforms with specified SNRs. In Part 2, you will study the differences between the Continuous Fourier Transform (CFT) and the Discrete Fourier Transform (DFT). In Part 3, you will synthesize AM and FM bandpass signals and analyze their spectra. In Part 4, you will capture and analyze the spectra of a segment of any piece of music that you like.

Software

Matlab: https://www.mathworks.com/help/pdf_doc/matlab/getstart.pdf

Part 1: Digital synthesis of arbitrary waveforms with specified SNR

Background

SNR in dB = 10 log₁₀ (s_s²/s_n²); where s_s²:signal variance, s_n²: noise variance
Given signal, s(t), find s_s²
Compute required s_n²
Generate noise signal, n(t) = s_nN(0,1), where N(0,1) is a Normally (Gaussian) distributed random variable with Zero mean and Unit variance
Message signal with desired SNR, m(t) = s(t) + n(t)

Procedure Overview

Synthesize a 1 second A-Sharp sinusoidal tone (466.16 Hz) sampled at 8 kHz.
Plot this waveform; observe and listen using the sound command in Matlab.
Corrupt this signal with a Gaussian noise source to get an SNR of 10 dB. Observe, listen and save waveform as before.
Synthesize 1 cycle of the noisy waveform.
Vary the frequency, amplitude.

Repeat experiment with various SNRs.

Experiment with other waveforms:

s(t) = A_c[1 + cos(2pf_mt)]cos(2pf_ct) with varying A_c, f_mand f_c.
s(t) = A_ccos[2pf_ct + bsin(2pf_mt)] with varying A_c, b, f_mand f_c.

Example Matlab Code

%ECOMMS Lab Project 1 Example
%S. Mandayam, Rowan University
%This program generates a 1-second duration
%Asharp signal (466.16 Hz) with a specified SNR

%Specify SNR
snr=10;

%Generate Asharp signal
t=[0:1/8e3:1.0]';
s = 0.5*sin(2*pi*466.16*t);
sound(s);

%Compute signal variance
var_s = cov(s);

%Calculate required noise variance
var_noise=var_s/(10^(snr/10));

%Generate noise
n=sqrt(var_noise)*randn(length(s),1);
sound(n);

%Add signal to noise and generate message
m=s+n;
sound(m);

Part 2: Comparison between CFT and DFT (FFT)

Consider the signal shown in Figure 1:

Figure 1: Time-domain signal

Model the signal as a continuous-time function and plot.
Obtain, analytically, the CFT of the signal, and plot.
Based on your observations of the frequency components of the signal, determine the maximum sampling period/minimum sampling frequency that will allow reconstruction of the continuous-time signal.
Plot the DFT magnitude spectrum using Matlab's fft function.
Attempt to reconstruct the original continuous-time function from its samples, either by:

Convolving the time domain samples with appropriate Sinc function (difficult), or
Windowing the Fourier transform and taking the inverse Fourier transform (easier).

Comment on your results.

What is the maximum sampling period/minimum sampling frequency that will allow reconstruction of the discrete-time signal from its DFT, which adequately represents the original continuous-time signal? Show plots of the discrete-time signals, the corresponding DFTs and reconstructions, for sampling intervals at, above and below this maximum allowed sampling period.

Part 3: Spectral Analysis of AM and FM Signals

Synthesize a bandpass AM signal, s(t) = A_c[1 + A_mcos(2pf_mt)]cos(2pf_ct) where f_m = 5 kHz and f_c = 25 kHz.
Obtain and plot the spectral components of this signal using Matlab's fft function.
Add noise to the RF signal, observe the signal in the time and spectral domains.
Experiment with various f_ms, f_cs and SNRs.
Synthesize a bandpass FM signal, s(t) = A_ccos[2pf_ct + b_fA_msin(2pf_mt)] whereb_f = Frequency Modulation Index (choose initially = 10).
Repeat earlier experiments.

Part 4: Spectral Analysis of Music Signal

Legally, download a segment of your favorite piece of music, import into Matlab and listen. Compute its spectrum in Matlab, observe.

Click here for required lab project report format.

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