Electrical Communications Systems

Course No. ECE.09.433

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Laboratory Project 1

Waveform Synthesis and Spectral Analysis

Objectives

This project has 4 parts. In Part 1, you will:

• Generate arbitrary waveforms with specified SNR's using software;
• Synthesize this waveform using an arbitrary waveform generator.

In Part 2, you will study the differences between the Continuous Fourier Transform (CFT) and the Discrete Fourier Transform (DFT) (this is homework!). 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.

Equipment and Software

• You will be synthesizing waveforms in Matlab and creating their corresponding electrical signals using the Arbitrary Waveform Generator. In order to do this, you will need the Keysight IO Libraries Suite. Instructions have been created by Yang Qi with sample Matlab code.
• You will be capturing signals from the Oscilloscope/Spectrum Analyzer and will be analyzing them in Matlab.

Part 1: Digital synthesis of arbitrary waveforms with specified SNR

Background

Pre-Lab Lecture

• 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.
• Download the original and noisy signals to the HP33120A Arbitrary Function Generator.
• Vary the frequency and amplitude.
• Observe waveform using the Oscilloscope.
• 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_m and f_c.
• s(t) = A_ccos[2pf_ct + bsin(2pf_mt)] with varying A_c, b, f_m and 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)
(Uses Matlab only) (This is Homework!)

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
(Uses Matlab, Oscilloscope, Function Generator and Spectrum Analyzer)

• 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.
• The spectrum analyzer/ FFT module.
• 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_f A_msin(2pf_mt)] whereb_f = Frequency Modulation Index (choose initially = 10).
• Repeat earlier experiments.

Part 4: Spectral Analysis of Music Signal
(Uses Matlab, Oscilloscope, VCR,  and  Spectrum Analyzer/FFT Module)

Legally, download a segment of your favorite piece of music, import into Matlab and listen. Compute its spectrum in Matlab, observe. Compare with the spectrum displayed in the Spectrum Analyzer.

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