THE CITADEL THE MILITARY COLLEGE OF SOUTH CAROLINA. Department of Electrical and Computer Engineering. ELEC 423 Digital Signal Processing

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1 THE CITADEL THE MILITARY COLLEGE OF SOUTH CAROLINA Department of Electrical and Computer Engineering ELEC 423 Digital Signal Processing Project 2 Due date: November 12 th, 2013 I) Introduction In ELEC 423, we have been developing mathematical tools to represent discrete-time signals and to understand how signals interact with discrete-time systems. Of interest is one class of signals that have been extensively studied in real applications called audio signals. Audio signals represent longitudinal waves propagating through air that the human ears perceive as sound. Audio signal processing is a fascinating area of study. It has very many practical applications, especially in the consumer electronics industry. Nowadays, the internet is full of software for various different inds of audio signal processing. The goal of this project is to explore the theories of sampling and reconstruction in the context of audio signals. II) Part I Project Description A pure tone is the sound associated with a perfect sinusoidal signal of the form x ( t ) = A sin( Ω t 0 ). It refers to a single sinusoidal signal of a certain constant frequency. The amplitude A is proportional to the intensity or the loudness of the sound. The frequency Ω 0 is proportional to the pitch of the sound. 2π Let us sample x (t) above at a sampling frequency Ω s = rad/s to obtain the discretetime (DT) signal x[ n] = x( nt s Ts ) = sin( Ω0nTs ). 1. Generating pure tones Write a MATLAB user-defined function called sinusoid.m with the following function definition: function [x t] = sinusoid(omega_0,dur,omega_s) that generates a vector x containing the samples of a CT sinusoid of frequency omega_0 for a duration of dur seconds, sampled at the sampling frequency of omega_s. The function should also output a vector t, containing the time-instants at which the samples are taen. Hence, the command max(t) should return the total duration dur. [Hint: Figure out how many samples nmax should the vector x contain. You might need to use the function fix() to truncate a real number to get an integer. Then, create a vector n = [0:nmax], so that t = n*t s contains the nmax time samples of the interval 0 t < dur seconds. Then create a vector x which contains the samples of x(t) at the time samples in t.] 1

2 2. Generating musical notes There are 12 musical notes in an octave C, C#/D b, D, D#/E b, E, F, F#/G b, G, G#/A b, A, A#/B b, B. They are calibrated so that the frequency of the note A above middle C is 440 Hz (referred to as A440 ). In an equally tempered scale, the frequency of each subsequent note is 2 1/12 times the frequency of the preceding notes. Thus, the frequency of middle C is /12 = Hz and the frequency of the note G is /12 Hz. After the note B, comes the note C of the higher octave, and all the notes repeat thereafter in the same order in the higher octave. So, 110 Hz, 220 Hz, 440 Hz, and 880 Hz are all A at different octaves. The frequency of the note C of the higher octave is given by /12 = = Hz, doubling the frequency of C of the middle octave. Most pianos are tuned to the equally tempered scale. Though, some traditional musicians prefer to play the diatonic scale, in which the ratios of the frequencies of successive notes are not exactly equal. But, the advantage of using the equally tempered scale is that one can transpose a song to a different note pretty easily just by up-converting or down-converting the frequencies of all notes by a fixed factor. 2.1 Using the function sinusoid.m that you wrote in 2.1.1, generate a vector x containing samples of a pure tone of the note D of the middle octave of duration 3 seconds. Use a sampling frequency of Ω s = 2π (8192) rad/s. Call this signal x[n]. Remember to convert the frequencies from Hz (cycles/s) to rad/s. 2.2 Use the command sound(x,1/ts) to listen to the note that you have produced, where T s = 1/8192 is the sampling period in seconds. You may wish to chec using a musical instrument that this sound is indeed a note D. 2.3 Now, generate a 1-second long sinusoid of frequency Ω0 = 2π (1000) rad/s. Repeat step 2.2 for this signal. What note does this sound best approximate? 3. Frequency characteristics of signals Note from your plots above that the command plot(t,x)displays a CT signal given the samples in x, using straight lines to interpolate between sample values. While this interpolation is not generally equal to the band-limited reconstruction that follows from the sampling theorem, it can often be a good approximation. To compute samples of the CTFT of the band-limited reconstruction x r (t), use the following MATLAB function ctfts: function [X f] = ctfts(x, Ts) % - CTFTS calculates the CTFT of a periodic signal x(t) which is % reconstructed from the samples in the vector x using ideal % band-limited interpolation. % - The vector x contains samples of x(t)over an integer number % of periods, and Ts contains the sampling period. % - The vector X contains the area of the impulses at the % frequency values stored in the vector f. % - This function maes use of the relationship between the CTFT % of x(t) and the DTFT of its sample x[n] and the DTFS of x[n]. % N = length(x); X = fftshift(fft(x,n))*(2*pi/n); f = linspace(-1,1/n,n)/(2*t); 2

3 Use the function ctfts to calculate the CTFT of the reconstructed signal x r (t) for the 1000-Hz signal generated in 2.4 of above. Plot the magnitude of X versus f. Is X nonzero at the proper frequency values? Note that the above function plots the CTFT as a function of frequency f in Hz. Also, plot the phase of X as a function of f. 4. Combination of musical notes If signals corresponding to more than one different pure tones are added, i.e., x 1( t) + x2 ( t) + x3 ( t) +LL, where each x i (t) is a pure tone of a different frequency, the effect is that of notes of these different frequencies being played together, such as hitting multiple piano eys at once, or playing the violin bow on two strings at once. 4.1 The notes C, E, and G, when played together simultaneously constitute the C-major chord. Generate a 1-second long signal corresponding to the C-major chord of the middle octave. Use sampling frequency Ω s = 2π (8192) rad/s. Listen to the sound produced. 4.2 If the signal produced in (4.1) above were of infinite duration, would it be periodic? If yes, what is the period? If not, explain why not. 4.3 With the approximations 2 1/3 1.25, and 2 7/12 1.5, use ctfts.m to compute the CTFT of this signal. Plot its magnitude and phase as a function of frequency. Describe and explain any prominent features. 5. Fourier series representation of periodic signals Consider a periodic square wave x(t) of fundamental period T 0, such that 1, 0 t < T0 / 2 x( t) =. 1, T0 t < T0 Using the trigonometric CT Fourier series representation of a periodic square wave, developed in project 1, it can be shown analytically that x(t) can be expressed as: 2πt x( t) = a sin, = 1 T0 4 where a = 0 for even, and a = for odd. π 5.1 Generate a 3-second long signal xd_sq containing an approximation of a square wave of period T 0 corresponding to note D (see part 2 above), by adding the first 10 odd harmonics, i.e., = 1, 3, 5,., 19. Also, generate a 3-second long pure tone xd of note D (as you did in part 2). Use a sampling frequency of Ω s = 2π (8192) rad/s for both signals. 5.2 Play these two signals using the command sound(x,1/ts) command. What difference do you notice between the two? Do these two sounds have the same pitch? Note: When two musical instruments play the same note, what maes the two notes sound different are the different set of values of the CTFS coefficients of the two signals (of the same fundamental frequency) generated by the instruments. Though a pure tone is literally the purest tone (CTFS a is non-zero only for ω = ±ω0 ) that one may generate on one single frequency, it is really the higher harmonics (CTFS coefficients a for > 1) that contribute to the richness of a musical note produced by an instrument. 3

4 6. Generating Music The following is a set of step-by-step instructions on how to generate music using MATLAB. 6.1 Generating two pure tones in succession. Generate two vectors x1 and x2 in MATLAB corresponding to two 0.5-second long pure tones of two different notes, C and D. Again, use the default sampling rate Ω s = 2π (8192) rad/s. Let us append these two row vectors one after the other, by using the command x = [x1 x2]. Play the sound for the signal x, you will hear a 1-second long sound, of which the first half-second will play the note C, and the second half will play D. 6.2 Generating a note played on an instrument. As mentioned before, what determines the quality of a sound generated by an instrument, are the CTFS coefficients of the periodic signal generated by the instrument when it plays a note. Now, using the same idea as in part 5, generate a vector xi1 corresponding to one 0.5-second long tone, on the note C. Use the first five integer sine-harmonics, with 5 { a } = {1, 0.5, 0.5, 0.3, 0.2} and the default sampling frequency of Ω = 2π (8192) = 1 s rad/s. In other words, xi1 should contain the samples of the signal: 5 x ( t) = a sin( Ω t), = 1 where Ω0 = 2π (261.63) rad/s is the frequency of the C note of the middle octave. Generate another similar 0.5-second long vector xi2 corresponding to the D note, and append the two notes by using xi = [xi1 xi2]. Play the sound for the signal xi. You should hear a 1- second long sound, of which the first half-second will play the note C, and the second half will play the note D, in the instrument described by the a values above. 6.3 Playing a song Now, quite evidently, if you eep appending vectors of samples of different notes generated by a piano, you can create and play a song. To play a silence (to play nothing) for a certain time duration, you may just generate samples of an all-zero signal. Remember that, the command sound(x, 1/Ts) assumes that the values of the samples in x lie in [-1,1]. Write MATLAB codes that play a few lines of your favorite songs using the above approximation of the piano. 0 4

5 III) Part II Project Description This project deals with the implementation of a sliding window DFT spectrum analysis system. To create a scenario that has some element of challenge, imagine a system whose job is to distinguish various tones. An example application might be a touch-tone telephone in which each tone represents a digit. This is not how a touch-tone phone actually wors, but it maes a reasonable example for the sliding DFT. Since the primary objective of this project is the implementation of a system to estimate frequencies, a data file containing a mystery signal is provided to test your final processing algorithm. 1 Sliding Window DFT Nonstationary signals are characterized by changing features in their frequency content with respect to time. Therefore, the Fourier analysis of such signals needs to be localized and also time-dependent. The resulting analysis methods are called time-frequency distributions. In this project, we will consider an elementary time-frequency distribution based on FFT. This elementary operation is called the STFT (short-time Fourier transform). Specific reading related to this subject can be found in section 3.10 The Spectrogram, of the textboo. The spectrogram is defined as a short-time Fourier transform of a windowed segment of the signal x[n]. X n ( e jω ) = n m= n L+ 1 x[ m] w[ n m] e The window w[ ] determines how much of the signal will be used in the analysis and controls the frequency resolution of the Fourier analysis. The parameter n denotes the reference position of the window on the signal. The implementation of the equation above requires a loop to compute X ( e jω n ) for a succession of window positions n. At each window position, the FFT is computed for an L-point bloc of the signal. Usually, n is incremented by a fraction (25 to 50%) of the window length L between successive FFTs. In other words, there is usually a 25 to 50% overlap of the data in the present analysis window and the previous analysis window. 2 Frequency-coded Input Signal The input signal provided must be analyzed to determine its frequency content versus time. The following description of the input signal helps you become familiar with the characteristics of the three test signals and the mystery signal. The three test signals are provided for pre-testing your system. The input signal is nown to be composed of five successive short-duration sinusoids of constant frequency. The duration of each sinusoid is not necessarily the same, but the length of each is usually between 15 and 20 ms. The desired signal encodes a sequence of digits by using different frequencies to represent individual digits, according to the following table: jmω

6 Digit Freq.(Hz) Thus, the signal changes frequency to one of the values in the table roughly every 15 to 20 ms. For example, the digit sequence would consist of a sinusoid of frequency 3750 Hz, followed by one of 1750 Hz, then 2250 Hz, 750 Hz, and finally 4750 Hz. To avoid instantaneous changes in frequency, the step changes in frequency were filtered through a low-pass filter to provide some smoothing of the frequency versus time. The sampling rate is F S = 10 Hz, and for this project, you can assume that the sampling was carried out perfectly-no aliasing. In addition, some of the input signals contain the second component, which is an interference signal that is quite stronger than the frequency-coded signal component. This interference is a narrowband signal (i.e., sinusoid) whose frequency is changing continuously with time. Its pea amplitude is about 10 times stronger than the coded signal. You will have to determine the value of this amplitude and the frequency variation of this signal versus time. However, there is no need to remove this interference prior to performing the slidingwindow DFT. 3 Implementation The primary tas in this project is to design and implement the sliding-window DFT system so that it will reliably extract the coded information in the given input signal described earlier. Since the method of analysis to be implemented is the sliding-window DFT, you are to pic the parameters of the system: window length, window type, FFT length, segment overlap, and so on. For the purpose of implementation, the total signal length can be assumed to be about 100 ms so that the frequency-coded input signal contains 5 digits. Your system must be able to identify these 5 digits. This can be done in two ways: a) Visually inspect the DFT plot for each windowed segment of the input signal to extract the information (locations of peas and the corresponding digits), or b) Design an automatic pea-picing algorithm to extract the code automatically. The algorithm should wor for any combination of input digits. Remember that the objective of your design is to minimize the amount of computation as much as possible because this means a reduction in cost. 4 System Evaluation To demonstrate that your system wors correctly, you must process the three nown test signals and the mystery signal. The first test signal has the [ ] digit sequence, the second the [ ] sequence, and the third the [ ] sequence. 2.5 Project Report No formal project report is required except for a summary sheet of your algorithm including bloc diagram and your choices of parameters used. You must also submit your MATLAB code both electronically and in paper form. In addition, to demonstrate that your algorithm wors, it should automatically output the desired digit sequence when executed.

7 IV) Project Report 1) Submit your MATLAB code in both paper and electronic forms. If you want to explain the rationale of your program, you may want to include comments in your MATLAB code. 2) Electronic submission must be done via to If possible, all files should be zipped together into one big file (using a utility program such as WINZIP) and attached to your . Use the following naming convention for your zip file: Project_2_ELEC423_ your name.zip 5) Your MATALB programs must begin with the following three lines of comment: % program_name.m % State the purpose of the program % Written by your_name on due_date 6) COMPLETION OF ALL ASSIGNED PROJECTS AND WRITEUPS IS A REQUIREMENT TO PASS THE COURSE. So, start woring on the project early!!!! 7