Eee312_new_lab01_rev_oct2018_2.pdf

Lab Sheet 1 Analog to Digital Conversion (ADC) and Digital to Analog Conversion (DAC) Fundamentals 

Prerequisites The recommended prerequisites for this lab are as follows…

1. A fundamental knowledge on MATLAB 2. A theoretical knowledge on sampling,reconstruction and quantization.

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Outcomes

After completing this lab, the students will 1. Understand signal sampling and reconstruction and their implementation using MATLAB. 2. Understand quantization and its implementation using MATLAB.

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Outlines of This Lab 1. 2. 3. 4. 5. 6.

Lab Session 1.1: Sampling of Continuous Time (CT) signal Lab Session 1.2: Reconstruction of Discrete Time (DT) signal to generate CT signal Lab Session 1.3: Observation of effect of aliasing Lab Session 1.4: Quantization Lab Session 1.5: Simple ADC and DAC Systems Home work

1.1 Sampling of CT signal 1.1.1 Introduction: Signals available in our environment are mostly analog. As a result we cannot use them for digital signal processing since digital signal processing (DSP) deals with discrete time signal. This requires the sampling of an analog signal and then quantizing the amplitude of the sampled signal.

1.1.2 Sampling: Sampling is a process of converting a continuous time signal into a discrete time signal. To convert a continuous signal into a discrete signal, samples of the signal are taken periodically, at regular intervals.

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If y (t ) is a continuous time signal, the corresponding discrete signal will be y(nTs) where n  1, 2,3, 4,.... and Ts is the sampling interval. We can also write the discrete signal as y (n) implying the sampling period of Ts.

1.1.2 Sampling Frequency: Sampling frequency is the number of samples taken per second by the sample block of ADC. 1 The sampling frequency and Ts are related by f S  . Ts

1.1.3 The Nyquist–Shannon Sampling Theorem: The Nyquist–Shannon sampling theorem is a fundamental result in the field of information theory, in particular telecommunications and signal processing. The theorem is commonly called Shannon's sampling theorem. The theorem states that exact reconstruction of a continuous-time baseband signal from its samples is possible if the signal is bandlimited and the sampling frequency is greater than twice the signal bandwidth. The theorem also leads to an effective reconstruction formula.

1.1.4 MATLAB Example Example 1.1: Consider an analog signal x a (t )  cos(20t ) , 0  t  1 . It is sampled at Ts = 0.01, 0.05 and 0.1sec intervals to obtain x(n). For each Ts, plot x(n). Comment on your results. Note: To represent a CT signal in MATLAB, x(t) is generated using time interval of 0.001s and plotting it without any line style.

Solution: T=1; t=0:0.001:T; xa=cos(20*pi*t); Ts=0.01; N1=round(T/Ts); n1=0:N1-1; x1=cos(20*pi*n1*Ts); subplot(3,1,1); plot(t,xa,n1*Ts,x1,'o');axis([0,1,-1.1,1.1]); ylabel('x1(n)'); title('Sampling of x_{a}(t)using Ts=0.01'); Ts=0.05; N2=round(T/Ts); n2=0:N2; x2=cos(20*pi*n2*Ts); subplot(3,1,2); plot(t,xa,n2*Ts,x2,'o'); axis([0,1,-1.1,1.1]); ylabel('x2(n)'); title('Sampling of x_{a}(t)using Ts=0.05'); Ts=0.1; N3=round(T/Ts); n3=0:N3; x3=cos(20*pi*n3*Ts);subplot(3,1,3); plot(t,xa,n3*Ts,x3,'o'); axis([0,1,-1.1,1.1]); ylabel('x3(n)');title('Sampling of x_{a}(t)using Ts=0.1');

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Output: Sampling of x a(t)using Ts=0.01

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Sampling of x a(t)using Ts=0.05

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Sampling of x a(t)using Ts=0.1

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1.2 Reconstruction of CT signal From the sampling theorem and the preceding example, it is clear that if we sample band-limited xa(t) above its Nyquist rate, then we can reconstruct xa(t) from its samples x(n). This reconstruction can be thought of as a 2-step process: (i) First the samples are converted into a weighted impulse train.

(ii)

Then the impulse train is filtered by an ideal analog lowpass filter band-limited to the [−Fs/2, Fs/2] band.

This two-step procedure can be described mathematically using an interpolating formula

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Where

is an interpolating function. We observe that this ideal interpolation is not practically feasible because the entire system is non causal and hence not realizable.

1.2.1 MATLAB Example Example 1.2: Reconstruct the analog signal ya(t) from the samples x(n) (from example 1.1) by means of the sinc interpolation. (Use t=0.001). Estimate the frequency in ya(t) from your plot. Comment on your results.

Solution: The following block of codes implement the sinc interpolation as shown in the equation for reconstruction for Ts=0.01 sec. This code needs to be appended to the end of block of codes in Example 3.1 so that the sampled signals x1, x2 and x3 can be used to reconstruct the corresponding analog signals. You need to change the variables Ts, n1, x1 and xa1 to their corresponding variable names for Ts= 0.05 and 0.1 sec (i.e., change n1 to n2, x1 to x2 etc. for Ts=0.05). %% For Ts= 0.01 Ts=0.01; xa1 = zeros(1,length(t)); for i=1:length(x1) g=sinc((t - (i-1)*Ts)/Ts); xa1 = xa1+x1(i)*g; end %% For Ts= 0.05 Ts=0.05; % fill in codes %% For Ts= 0. 1 Ts=0.1; % fill in codes figure subplot(3,1,1); plot(t,xa1) axis([0,1,-1.1,1.1]); ylabel('x_{a}(t)'); title('Reconstruction of x_{a}(t) when Ts=0.01 '); subplot(3,1,2); plot(t,xa2) title('Reconstruction of x_{a}(t) when Ts=0.05 ');

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Output: Reconstruction of x a(t) when Ts=0.01

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Reconstruction of x a(t) when Ts=0.05

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Reconstruction of x a(t) when Ts=0.1

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Comments: From the plots, it is clear that reconstruction from samples at Ts=0.01 and 0.05 depicts the original frequency, but the reconstruction for Ts = 0.1 shows the original frequency aliased to zero.

1.3 Observation of effect of aliasing: Example 3.3: Plot three functions on same plot as follows:

(i) x1  cos 6t (ii) x 2  cos 14t (iii) x3  cos 26t Use t = nT, T=0.001sec and n =500 Now sample (i) at Ts = 0.1sec. Plot this on the same plot also. What do the results indicate? Explain.

Solution : T=0.001; n=0:500; x1=cos(2*pi*3*n*T); x2=cos(2*pi*7*n*T); x3=cos(2*pi*13*n*T); plot(n*T,x1); hold on; plot(n*T,x2,'r'); plot(n*T,x3,'g'); Ts=0.1; N=round(1/Ts); n1=0:N/2; x1=cos(2*pi*3*n1*Ts); plot(n1*Ts,x1,'o');

Repeat for x2 and x3 and plot with different line markers.

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Output: 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1

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1.4 Quantization 1.4.1 Theoretical background: In digital signal processing, quantization is the process of approximating a continuous range of values (or a very large set of possible discrete values) by a relatively small set of discrete symbols or integer values. Quantization is the representation of the sampled values of amplitude by a finite set of levels. It converts a continuous-amplitude sample to a discrete-amplitude sample.

Here, b = number of bits assigned to quantize a signal at L = 2b and Δ = 6|Page

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For any sampled amplitude of x, its quantized value can be determined by finding its location in terms of Δ first as h = round ( ) Then the quantized value can be given by Xq = hΔ + Xmin.

1.4.2 Custom function for quantization: function y=uquant(x,N) %x= original sequence %N=Number of steps maxval=max(max(x)); minval=min(min(x)); stepsize=(maxval-minval)/(N-1); h=round((x-minval)/stepsize); y=h*stepsize+minval;

Example 1.4: Use the function uquant( ) to quantize a vector: x= {1 2 -3 -4 5 10 12 13 16 11 0 -5 7 9 10 16} at a level (a) 2 (b) 4 (c) 8 (d) 16

Solution: x=[1 2 -3 -4 5 10 12 13 16 11 0 -5 7 9 10 16 ]; y1=uquant(x,2) y2=uquant(x,4) y3=uquant(x,8) y4=uquant(x,16)

Output The data output is : x= 1 2 -3 -4 5 10 y1 = -5 -5 -5 -5 -5 16 y2 = 2 2 -5 -5 2 9 y3 = 1 1 -2 -5 4 10 y4 = Columns 1 through 9 0.6000 2.0000 -3.6000 Columns 10 through 16 10.4000 0.6000 -5.0000

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You can also find quantization error or noise (e.g., for y1) by computing quant_noise1= x-y1; stem(quant_noise1);

After finding the quantization noise or error for all quantized outputs (y1 to y4), stem them in the same figure in different sub-plots. You will notice that the quantization noise or errors reduce as b is increased. Explain why this happens. 7|Page

1.5 Implementation of Simple ADC and DAC 1.5.1 Introduction In this session we‟ll put together our codes for sampling, quantization and reconstruction to implement a simple ADC for generating quantized samples followed by a DAC to regenerate the analog samples for various sampling rates and quantization levels. This will give us a system level understanding of the functions of ADC and DAC and the fidelity of signal waveform as it is subjected to the processes of ADC and DAC. An ADC comprises sampling followed by quantization. The DAC consists of sinc interpolation of the output of ADC. We‟ll use the code from Example 3.1 to implement our ADC and DAC. The following example illustrates the implementation. You can change b and Ts to see the effect of ADC and DAC on the signal quality.

1.5.2 MATLAB Example Example 1.5: clear all close all %% Parameter file: Ts=0.01; %Sampling interval b= 2; % Number of bits in Quantizer T=1; %Time duration for analog signal F= 10; %Frequency of analog signal %% ADC: % Generation of analog signal: t=0:0.001:T; xa=cos(20*pi*t); % Sampling N1=round(T/Ts); n1=0:N1-1; x1=cos(20*pi*n1*Ts); % Quantization: L= 2^b; %number of levels y1= uquant(x1,L); %% DAC: xa1 = zeros(1,length(t)); for i=1:length(y1) g=sinc((t - (i-1)*Ts)/Ts); xa1 = xa1+y1(i)*g; end %% Plots: figure plot(t,xa) hold on plot(t,xa1,'r') ylabel('Signal level'); legend('Original Signal', 'Reconstructed signal') title('ADC and DAC operation');

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Home Work: HW1.01: Given a sinusoid waveform with a frequency of 100 Hz, x(t)  4.5sin  2 100t  sampled at 8000Hz. Write a MATLAB program to quantize x(t) using 4 bits to obtain and plot the quantized signal, assuming the dynamic range is between -5 to 5 volts. HW1.02: An analog signal xa(t) =sin(1000t) is sampled using the following sampling intervals. In each case, plot the DT signals and their corresponding reconstructed signals in the same figure. (a) Ts = 0.1ms (b) Ts = 1ms (c) Ts = 0.01ms

Authorship: This lab sheet was prepared by B.K.M. Mizanur Rahman, Assistant Professor of EEE Department of UIU. It was reviewed and revised by Dr. Raqibul Mostafa, Professor of EEE Department in Fall 2017.

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Optional work materials

Lab Session 1.4:Effect of sampling on sound 1.4.1: USING sound( ) TO DEMONSTRATE THE ALlASING EFFECT MATLAB has a function sound() that can convert a discrete signal sequence into continuoustime sounds. We can use it to demonstrate the effect of aliasing very easily.

Example 1.4: Introduce the “chirp” phenomena where the instantaneous frequency of some special signal will increase over time. Consider the signal

1   x(t )  sin  o t   t 2  2   Set o = 2(3000) rad/sec and  = 2000 rad/sec2.

1. Store in the vector x the samples of the chirp signal on the interval 0t1, and let T is the same value as above. 2. Use sound(x,fs) to play the chirp signal contained in x. Can you explain what you just heard? 3. Can you predict the time at which the played signal has zero (or very low) frequency? Use a longer x sequence to verify your prediction.

The MATLAB code : f=3000; %frequency of the sine wave fs=8192; %sampling frequency Ts=1/fs; %sampling time Ns=512; %512 points t= [0:Ts:Ts*20*(fs-1)]; %50/8000~6.25ms x=sin(2*pi*f*t+0.5*2000*t.^2);%sine function for 50 samples sound(x,fs) %play the wave

Task: (i) Change fs=1000, 2000, 3000, 5000,6000, 8000 respectively and repeat the above code. Can you explain what you just heard in each case? Why the sound is distorted below a certain frequency? Try to detect the sampling frequency below which the sound you have just heard is distorted.

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Example 1.5: Playing practical sound:

Steps: (i) Load an audio file: [y fs ]=audioread('audioclip1.wav'); % Place the file 'audioclip1.wav' in the current folder [y fs ]=auread(„'audioclip1.wav‟) returns the sample rate (fs) in Hertz and the number of bits per sample (bits) used to encode the data in the file.

(ii) plot the function : >> plot(y) The following curve will be shown : 0.6

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(iii) Find the sample rate : >> fs Output: fs =16000 (iv) Play the sound : >> sound(y,fs) (vi) Find the size of the sound file : >> size(y) Output: ans = 410530 1 (v) Play sound at different sampling frequency: >> sound(y,8000) % hear the sound (vi) Play sound at different sampling frequency: >> sound(y,32000) % hear the sound

Comment on results of (v) and (vi)

Lab Session 1.6: Effect of quantization on image : Example 1.5:

Steps (i) Load an image : x=imread('fountainbw.tif'); % Make sure that the image is available 11 | P a g e

x=double(x); image(x) The following image will appear : 50 100 150 200 250 300 350 400 50

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(ii) Make the image gray : >> colormap(gray(256)) (iii) Make image size square >> axis('image')

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The figure will be gray colored. The image will be like the following:

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(iv) Quantize the image at different level and see the effect :

MATLAB code : y=imread('fountainbw.tif'); subplot(2,4,1); image(y) colormap(gray(256)); axis('image'); y=double(y); for b=1:7 N=2^b; J=uquant(y,N); subplot(2,4,b+1) image(J); colormap(gray(256)); axis('image'); end

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