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--- ecen_380_assignments [2015/10/16 10:02]
petersen
+++ ecen_380_assignments [2016/05/16 10:02] (current)
a_west [Chapter 1: LTI Systems]
@@ Line 1: / Line 1: @@
-===== ECEn 380 MATLAB Assignments =====
+====== ECEn 380 MATLAB Assignments ======
+Welcome to the 380 section for the ECEn MATLAB wiki. For a brief refresher on MATLAB, [[ matlab_guide | look here.]]
+The syntax for the problem numbers is such:
+<code>
+[Language][Class].[Book authors][Chapter].[Section].[Problem]
+</code>
+Thus for help on problem "MT380.UY6.8.1", for example, you can refer to section 6-8 in the textbook (page 279).
 <ifauth, @admin>
@@ Line 6: / Line 15: @@
 ^ Chapter ^  Assignments                                             ^  Course Topic                   ^ MATLAB Topic                                                                                                                                                                  ^
-| Ch 1    | [[ecen_380_assignments#MT380.UY1 | MT380.UY1 ]]           | Signals                         |                                                                                                                                                                               |
+| Ch 1    | [[ecen_380_assignments#Chapter 1: Signals | Signals ]]           |||
-| :::     | [[ecen_380_assignments#MT380.UY1.2.1 | MT380.UY1.2.1 ]]   | Signal Transformations          | [[matlab_guide#Array Operations| Array Operations]],[[matlab_guide#Graph Function| plot]] ,[[matlab_guide#Annotation| Graph Annotation]] ,[[matlab_guide#Figures| Figures]]   |
+| :::     | [[ecen_380_assignments#Signal Transformations | MT380.UY1.2.1 ]]   | Signal Transformations          | [[matlab_guide#Array Operations| Array Operations]],[[matlab_guide#Graph Function| plot]] ,[[matlab_guide#Annotation| Graph Annotation]] ,[[matlab_guide#Figures| Figures]]   |
-| :::     | [[ecen_380_assignments#MT380.UY1.5.1 | MT380.UY1.5.1 ]]   | Signal Power and Energy         |                                                                                                                                                                sum (used to evaluate numerical integrals)           |
+| :::     | [[ecen_380_assignments#Signal Power and Energy | MT380.UY1.5.1 ]]   | Signal Power and Energy         |                                                                                                                                                                sum (used to evaluate numerical integrals)           |
-| Ch 2    | [[ecen_380_assignments#MT380.UY2 | MT380.UY2 ]]           | LTI Systems                     |                                                                                                                                                                               |
+| Ch 2    | [[ecen_380_assignments#Chapter 2: LTI Systems | Linear Time-Invariant (LTI) Systems ]]           |||
 | :::     | [[ecen_380_assignments#MT380.UY2.3.0 | MT380.UY2.3.0 ]]   | Convolution  Aid                |                                                                                                                                                                               |
 | :::     | [[ecen_380_assignments#MT380.UY2.3.1 | MT380.UY2.3.1 ]]   | Convolution with function files |  [[matlab_guide#Function Files| Function Files]], [[matlab_guide#For-loop| For-loop]]                                                                                         |
@@ Line 16: / Line 25: @@
 | :::     | [[ecen_380_assignments#MT380.UY2.8.1 | MT380.UY2.8.1 ]]   | Imp.Resp. of 2nd Ord LCCDEs     |    [[matlab_guide#Roots| Roots]]                                                                                                                                              |
 | :::     | [[ecen_380_assignments#MT380.UY2.9.1 | MT380.UY2.9.1 ]]   | Car Suspension                  |    Review             																																						|
-| Ch 3    | [[ecen_380_assignments#MT380.UY3 | MT380.UY3 ]]           | Laplace Transform               |                                                                                                                                                                               |
+| Ch 3    | [[ecen_380_assignments#Chapter 3: Laplace Transform | Laplace Transform ]]           |||
 | :::     | [[ecen_380_assignments#MT380.UY3.2.1  | MT380.UY3.2.1 ]]  | Poles and Zeros                 |    [[matlab_guide#Mesh| Mesh]]                                     																										    |
-| Ch 4    | [[ecen_380_assignments#MT380.UY4 | MT380.UY4 ]]           | App. of the L-Transform         |                                                                                                                                                                               |
+| Ch 4    | [[ecen_380_assignments#Chapter 4: Applications of the Laplace Transform | Applications of the Laplace Transform ]]           |||
 | :::     | [[ecen_380_assignments#MT380.UY4.5.1  | MT380.UY4.5.1 ]]  | Op-Amp Circuits                 |   Review                                                                                                                                                                      |
-| Ch 5    | [[ecen_380_assignments#MT380.UY5 | MT380.UY5 ]]           | Fourier Analysis Techniques     |                                                                                                                                                                               |
+| Ch 5    | [[ecen_380_assignments#Chapter 5: Fourier Analysis Techniques | Fourier Analysis Techniques ]]           |||
 | :::     | [[ecen_380_assignments#MT380.UY5.4.1 | MT380.UY5.4.1 ]]   | Fourier Series Coefficients     |   [[matlab_guide#For-loop| For-loop]]                                                                                                                                         |
 | :::     | [[ecen_380_assignments#MT380.UY5.12.1 | MT380.UY5.12.1 ]] | Circuit Analysis with Fourier   |   Review                                                                                                                                                                      |
-| Ch 6    | [[ecen_380_assignments#MT380.UY6 | MT380.UY6 ]]           | Apps of the Fourier Trans.      |                                                                                                                                                                               |
+| Ch 6    | [[ecen_380_assignments#Chapter 6: Applications of the Fourier Transform | Applications of the Fourier Transform ]]           |||
 | :::     | [[ecen_380_assignments#MT380.UY6.8.1 | MT380.UY6.8.1 ]]   | Butterworth Filters             |  Review                                                                                                                                                                       |
 |    :::  | [[ecen_380_assignments#MT380.UY6.12.1 | MT380.UY6.12.1 ]] | Sampling Theorem                |  [[matlab_guide#fopen| fopen]],[[matlab_guide#fscanf| fscanf]],[[matlab_guide#fclose| fclose]], [[matlab_guide#interp| Interpolate]],[[matlab_guide#sound| sound]]            |
-| Ch 7    | [[ecen_380_assignments#MT380.UY7 | MT380.UY7 ]]           | Discrete-Time Signals and Syst  |                                                                                                                                                                               |
+| Ch 7    | [[ecen_380_assignments#Chapter 7: Discrete-Time Signals and Systems | Discrete-Time Signals and Systems ]]           |||
 | :::     | [[ecen_380_assignments#MT380.UY7.2.1 | MT380.UY7.2.1 ]]   | D-Time Signal Functions         |   rat                                                                                                                                                                         |
 | :::     | [[ecen_380_assignments#MT380.UY7.13.1 | MT380.UY7.13.1 ]] | D-Time Fourier Series           |  Review                                                                                                                                                                       |
+| :::     | [[ecen_380_assignments#MT380.UY7.15.1 | MT380.UY7.15.1 ]] | Discrete Fourier Transform      |  Review                                                                                                                                                                             |
+| Extras  | [[ecen_380_assignments#Music Harmonics | Music Harmonics ]] | FFT,Fourier Series Coeff        | FFT Plot
+====== Chapter 1: Signals ======
-===== Signal Transformation =====
+===== Signal Transformations =====
 ==== MT380.UY1.2.1 ====
-== Discription ==
+== Description ==
-<file>
+<file | MT380.UY1.2.1.txt >
  Signal Transformations
@@ Line 110: / Line 122: @@
 == Solution ==
-{{:380matlab:ch1:ul_1_2_signaltransformations.m|}}
+{{:380matlab:solutions:ch1:ul_1_2_signaltransformations.m|}}
 </ifauth>
@@ Line 179: / Line 191: @@
 == Solution ==
-{{:380matlab:ch1:uy_1_5_1_poweravg.m|}}
+{{:380matlab:solutions:ch1:uy_1_5_1_poweravg.m|}}
 </ifauth>
+====== Chapter 2: LTI Systems ======
 ===== Convolution Aid =====
@@ Line 335: / Line 348: @@
 == Solution ==
-{{:380matlab:ch2:h_t.m|}} \\
+{{:380matlab:solutions:ch2:h_t.m|}} \\
-{{:380matlab:ch2:x_t.m|}} \\
+{{:380matlab:solutions:ch2:x_t.m|}} \\
-{{:380matlab:ch2:mt380_uy2_3_1_convolutionff.m|}}  \\
+{{:380matlab:solutions:ch2:mt380_uy2_3_1_convolutionff.m|}}  \\
 </ifauth>
@@ Line 501: / Line 514: @@
 == Solution ==
-{{:380matlab:ch2:convolution.m|}}
+{{:380matlab:solutions:ch2:convolution.m|}}
 </ifauth>
@@ Line 561: / Line 574: @@
 == Solution ==
-{{:380matlab:ch2:uy2_5_1_convolutionproperties.m|}}
+{{:380matlab:solutions:ch2:uy2_5_1_convolutionproperties.m|}}
 </ifauth>
@@ Line 662: / Line 675: @@
 == Solution ==
-{{:380matlab:ch2:uy2_8_1_impresp2ndorder.m|}}
+{{:380matlab:solutions:ch2:uy2_8_1_impresp2ndorder.m|}}
 </ifauth>
@@ Line 696: / Line 709: @@
 == Solution ==
-{{:380matlab:ch2:uy2_9_1_eulermethod2ndorder.m|}}
+{{:380matlab:solutions:ch2:uy2_9_1_eulermethod2ndorder.m|}}
 </ifauth>
+====== Chapter 3: Laplace Transform ======
 ===== Laplace Transform Aid =====
@@ Line 794: / Line 809: @@
 == Solution ==
-{{:380matlab:ch3:uy3_1_1_sdomaingraph.m|}}
+{{:380matlab:solutions:ch3:uy3_1_1_sdomaingraph.m|}}
 </ifauth>
+====== Chapter 4: Applications of the Laplace Transform ======
 ===== Op-Amp Circuits =====
@@ Line 837: / Line 854: @@
 == Solution ==
-{{:380matlab:ch4:uy4_5_1_op_ampcircuit.m|}}
+{{:380matlab:solutions:ch4:uy4_5_1_op_ampcircuit.m|}}
 </ifauth>
+====== Chapter 5: Fourier Analysis Techniques ======
 ===== Fourier Series Coefficients =====
@@ Line 1010: / Line 1029: @@
 == Solution ==
-{{:380matlab:ch5:uy5_4_1_fourierseriescoefficients.m|}}
+{{:380matlab:solutions:ch5:uy5_4_1_fourierseriescoefficients.m|}}
 </ifauth>
@@ Line 1040: / Line 1059: @@
 == Solution ==
-{{:380matlab:ch5:uy5_12_1_circuitanalysisfouriertransform.m|}}
+{{:380matlab:solutions:ch5:uy5_12_1_circuitanalysisfouriertransform.m|}}
-{{:380matlab:ch5:h_t.m|}}
+{{:380matlab:solutions:ch5:h_t.m|}}
-{{:380matlab:ch5:x_t.m|}}
+{{:380matlab:solutions:ch5:x_t.m|}}
 </ifauth>
+====== Chapter 6: Applications of the Fourier Transform ======
 ===== Butterworth Filters =====
@@ Line 1053: / Line 1073: @@
  UY6_3_1_Active Filters
- Objective: model a second order butterworth filter using an approximated
+ Objective: Model a second order butterworth filter and test it by passing
- impulse response in order to see its effectiveness in filtering signals
+ signals at various frequencies through the filter. Since this is
+ a design problem, and you have already covered butterworth filters in
+ lab, less code will be given.
  functions: review roots
  Exercise: This exercise will be broken down into several parts
- part 1) in this first part you will create a low pass filter based on a
-       second order butterworth filter. Create it in a function file that
+ part 1) Creating the impulse response of the low pass filter.
-       takes in two parameters (corner frequency in Hz and time interval/step).
-       ex. functionName(fc , time_interval). The function file will return
+         You will model a second order lowpass filter using the butterworth
-       the impulse response h(t) in an array. Make the time length of the
+         design. The filter will be created in a separate function file
-       impulse response at least 10*tau; usually 5 tau is sufficient but
+         because you will use this filter in another MATLAB exercise. The function
-       let's be extra precise.
+         will take two arguments: the corner frequency in Hz and the
-       a) look at table 6-3 on page 285 in your book. This will help you
+         time-step (which we'll call "dt") used in creating the signals.
-          find the coefficients for your transfer function. Look at example
+         The function will
--10 in the book if you need more help. When designing this
+         return an array that represents the impulse response of the 2nd
-          transfer function, H(jw), make sure you have a dc gain of 1.
+         order lowpass filter. It is crucial that the dt used in
-       b) look at section 2-7.3 in your book to remember how to turn a
+         creating the filter is the same dt used to create the signal that
-          transfer function into a second order differential equation.
+         it is filtering. Below is the function declaration that is placed
-       c) review section 2-8 to transform your second order differential
+         in the function file that creates the filter.
-          equation into an impulse response, h(t).
-       If done correctly, you only need to pass the function file fc and
+             function [h] = h_t(fc,dt)
-       the time_interval and the function file will return an array
-       representing h(t).
+         Inside the function file you will need to convert the passed
- part 2) Now that you have constructed your filter it's time to test it.
+         in frequency in Hertz into radians per second:
-       a) create a low pass filter with an fc of 400 Hz using part 1 of
-          the assignment. Use an appropriate time interval. time_interval
+             wc = 2*pi*fc;
-          << tau. and time_interval << 1/largest(f), look at part b for
-          largest 4)
+         Section 6-8 in your text book shows how to create the frequency
-       b) create three different signals of the form x = 5*cos(2*pi*f*t).
+         response, H(jw), for a second order butterworth lowpass filter
-          the three signals will have different frequencies: f1 = 100, f2
+         using the specified frequency. For aid, refer to Table 6-3 (page 285)
-          = 400, f3 = 4000 Hz. When creating these signals, use the same
+         to obtain the pole locations and coefficients a1 and a2 that you
-          time_interval in part a) but have the time array last for 1 s.
+         will use to design your butterworth filter. Look at example 6-10
-       c) Convolve each signal with h(t) to simulate filtering.
+         in your book if you need more help. The frequency response should
-          y = x(t)*h(t), * denotes convolution.
+         look like the equation below.  (Remember that you can get the
-       d) plot h(t), and the three filtered signals.
+         frequency response from H(s) by substituting s = jw.)
+             H(jw) = wc^2/(s^2 + a1*wc*s + a2*wc^2)         (1)
+         The frequency response needs to be changed into its time equivalent
+         impulse response h(t). This will be done in two steps. In the first
+         step the transfer function needs to be changed into a second order
+         differential equation. If you have forgotten how to do this, refer
+         to section 2-7.3. The second order differential equation should
+         be of the form below.
+         d^2y/dt^2 + a1*dy/dt + a2*y(t) = b2*x(t)           (2)
+           - note, the coefficients a1 and a2 are not the same a1 and a2
+             coefficients in equation (1).
+         The second step is to transform the second order differential
+         equation into the impulse response h(t). Refer back to section
+-8. First calculate the poles (p1, and p2) as shown in equation
+.126 (pg 65) in your book.
+         p = roots([1 a1 a2]);
+           - note, a1 and a2 refer to the coefficients from equation (2).
+         Second, form the time array that will be used. The time array
+         will start from t =0s, increment by dt, and end at t = 10*tau.
+         Tau is the time constant obtained from the roots. The roots have
+         the basic form p = -sigma +/- j*wd where sigma represents the real
+         part and wd represents the complex part. Sigma is related to your
+         time constant, tau.
+           tau = 1/sigma                                 (3)
+         So figure out what your sigma is, and insert in your Matlab code:
+           tau = INSERT CODE HERE;   this is the code
+         Once tau is obtained, the time array can be created.
+              time parameters
+             time_beg = 0;
+             time_end = 10*tau
+             t = time_beg:dt:time_end - dt;  time array
+         The reason why we are using tau is because the impulse response
+         in non-finite.  However, its limit as t -> inf+ is 0.
+           lim h(t) = 0
+           t -> inf+
+         After about 5*tau the value of h(t) (or h(5*tau)) is almost zero
+         since the impulse response will decay exponentially. We are going
+         to t = 10*tau just to make sure that the behavior response is
+         completely captured. Also, since h(t) is causal, the time array
+         can start at t =0.
+         Now that we have the coefficients, roots, and the time array,
+         h(t) can be constructed using equation 2.136 in your book (pg
+).
+         To make things simple, below is a compilation of the above code
+         that you will put in your function file that creates the low pass
+         filter impulse response.
+             function [h] = h_t(fc,dt)
+             wc = 2*pi*fc;  the corner frequency in rads/s
+              a1_bwf and a2_bwf are coefficients obtained from table 6-3 in the book (pg 285).
+             a1_bwf = INSERT VALUE;
+             a2_bwf = INSERT VALUE;
+              a1 and a2 are coefficients associated with the 2nd order
+              differential equation.
+             a1 = INSERT CODE;
+             a2 = INSERT CODE;
+             r = roots([1, a1, a2]);  obtain the roots of the polynomial.
+              b2 is a coefficient according to equation 2.121 in the book it represents b2
+             b2 = wc^2;
+              Now let's set up the impulse response
+             tau = INSERT CODE;
+              time parameters
+             time_beg = 0;
+             time_end = 10*tau;
+             t = time_beg:dt:time_end - dt;  time array
+              impulse response
+              notice that the equation is simplifies because b1 = 0
+             h = INSERT CODE;
+ part 2) Now that you have constructed your filter, let's test it.
+         Copy the code below in a separate script and run the script.
+         The script will call your function file h_t.m to create the
+         lowpass filter and then pass three signals of amplitude 5 and
+         frequencies: 100,400,4000 Hz. The script will then plot the filtered
+         signals.
+             fc = 400;  corner frequency of filter(Hz)
+              time parameters
+             time_beg = 0;
+             time_end_x = 0.1;
+             dt = 1/100000;  time step
+             tx = time_beg:dt:time_end_x - dt;  time array for signals
+              impulse response of low pass filter
+             h = h_t(fc,dt);
+             th = time_beg:dt:length(h)*dt-dt;  time array for filter
+              frequencies for the two signals
+             f1 = 100;
+             f2 = 400;
+             f3 = 4000;
+              signals to be filtered
+             x1 = 5*cos(2*pi*f1*tx);
+             x2 = 5*cos(2*pi*f2*tx);
+             x3 = 5*cos(2*pi*f3*tx);
+              filtered signals
+             y1 = conv(x1,h)*dt;
+             y2 = conv(x2,h)*dt;
+             y3 = conv(x3,h)*dt;
+             time array for y
+             ty = time_beg:dt:length(y1)*dt-dt;
+             figure(1);
+              plots the impulse response
+             subplot(2,2,1);
+             plot(th,h);
+             xlabel('time (s)');
+             ylabel('amplitude');
+             title('impulse response h(t)');
+              plots the filtered signal with Hz = 100
+             subplot(2,2,2)
+             plot(ty,y1);
+             xlabel('time (s)');
+             ylabel('amplitude');
+             title('filtered signal with Hz = 100');
+              plots the filtered signal with Hz = 400
+             subplot(2,2,3)
+             plot(ty,y2);
+             xlabel('time (s)');
+             ylabel('amplitude');
+             title('filtered signal with Hz = 400');
+              plots the filtered signal with Hz = 4000
+             subplot(2,2,4)
+             plot(ty,y3);
+             xlabel('time (s)');
+             ylabel('amplitude');
+             title('filtered signal with Hz = 4000');
  part 3) You will now analyse the plots to verify that your filter is
          working.
-       a) you designed your filter with an fc of 400 Hz and a dc gain of
+       a) You designed your filter with an fc of 400 Hz and a dc gain of
 . Verify that the filtered signal with f1 = 100 Hz still has a max
           amplitude of 5.
@@ Line 1097: / Line 1280: @@
           signal has the same frequency as the corner frequency what should
           its amplitude approximately be?
-       d) Looking at the last signal with f3 = 4000 Hz ignore the first
+       d) Looking at the last signal with f3 = 4000 Hz, ignore the first
-          part of the filtered signal that has an amplitude twice the value
+          part of the filtered signal (the transient response) which has
-          that the rest of the signal has, remember that this is just an
+          an amplitude larger than the rest of the signal.
-          approximation. f3 is ten times the frequency of f2, and you are
+          f3 is ten times the frequency of f2, and you are
           using a second order filter. How much smaller should the
-          amplitude of the signal with a freq(f3) be compared to the
+          amplitude of the signal with frequency 4000 Hz be compare to the
-          signal(f2)?
+          signal with frequency 400 Hz?
 </file>
+== Solution Image ==
+{{:380matlab:ch6:uy6_3_1si_butterworthfilterssolutionimage.jpg?400|}}
 <ifauth @admin,@380ta>
 == Solution ==
-{{:380matlab:ch6:uy6_8_1_butterworthfilter.m|}} \\
+{{:380matlab:solutions:ch6:uy6_8_1_butterworthfilter.m|}} \\
 The function file below creates the impulse response of a second order lowpass filter. \\
-{{:380matlab:ch6:h_t.m|}}  \\
+{{:380matlab:solutions:ch6:h_t.m|}}  \\
 </ifauth>
@@ Line 1249: / Line 1438: @@
 == fft_plot function file ==
-{{:380matlab:ch6:fft_plot.m|}}
+{{:380matlab:solutions:ch6:fft_plot.m|}}
 == Original Music File ==
-{{:380matlab:ch6:pianofinal_5s.wav|}}
+{{:380matlab:solutions:ch6:pianofinal_5s.wav|}}
 == Music Files After Sampling ==
 This file was sampled at 1050 samples/sec, then filtered again with fc of 500 Hz.  \\
 You should be able to hear a low buz that is not in the original file \\
-{{:380matlab:ch6:pianos1nff.wav|}}
+{{:380matlab:solutions:ch6:pianos1nff.wav|}}
 This file was first filtered with fc of 500 Hz, sampled at 1050 samples/sec, then filtered again with fc of 500 Hz.  \\
 This file should sound similar to the original just quieter due to filtering \\
-{{:380matlab:ch6:pianos1ff.wav|}}
+{{:380matlab:solutions:ch6:pianos1ff.wav|}}
 This file was first filtered with fc of 500 Hz, sampled at 500 samples/sec, then filtered again with fc of 250 Hz.\\
 This file shows the effects of aliasing. \\
-{{:380matlab:ch6:pianos8ff.wav|}}
+{{:380matlab:solutions:ch6:pianos8ff.wav|}}
+====== Chapter 7: Discrete-Time Signals and Systems ======
 ===== D-Time Signal Functions =====
@@ Line 1319: / Line 1510: @@
 == Solution ==
-{{:380matlab:ch7:mt380_uy7_2_1_discrete_time_signal_functions.m|}}
+{{:380matlab:solutions:ch7:mt380_uy7_2_1_discrete_time_signal_functions.m|}}
-{{:380matlab:ch7:fundamentalperiod.m|}}
+{{:380matlab:solutions:ch7:fundamentalperiod.m|}}
 </ifauth>
@@ Line 1359: / Line 1550: @@
 == Solution ==
-{{:380matlab:ch7:mt380_uy7_13_1_discretetimefourierseries.m|}}
+{{:380matlab:solutions:ch7:mt380_uy7_13_1_discretetimefourierseries.m|}}
 </ifauth>
+===== Discrete Fourier Transform =====
+==== MT380.UY7.15.1 ====
+<file>
+ MT380 UY7 15 1 Discrete Fourier Transform
+Objective: The objective of the assignment is to create a function that
+           can implement the Discrete Fourier Transform and plot it.  We
+           have provided all of the code except for the actual
+           implementation of the DFT, but please go through the code
+           carefully and try to understand it.
+ Commands: Review
+ Exercise: You are given a file that represents a non-periodic signal. In
+           this exercise you will (1) window the signal, (2) apply the DFT
+           to the windowed signal, (3) and then plot the signals.
+           a) Begin by importing the signal file (file located below),
+              and storing its contents to a
+              variable. This variable will hold the data points of the
+              non-periodic signal.
+                 % this command will open the file. You will need to replace
+                 % filename with the name of the file.
+                 fileID = fopen('filename.txt');
+                 % the variable signal will contain the data points that represents the
+                 % signal.
+                 % the variable num_of_samples contains the number of data points in the
+                 % file.
+                 [original_signal,num_of_samples] = fscanf(fileID,'%e');
+                 % transpose the signal
+                 original_signal = original_signal';
+                 % Since you have what you need, you can close the file.
+                 fclose(fileID);
+                 % Below is the sample rate at which the original signal
+                 % was created, in samples/second.
+                   sample_rate = 1200;
+                 % plot the signal as a function of n(index)
+                 figure(1);
+                 subplot(2,2,1);
+                 plot(original_signal);
+                 title('original signal');
+               Notice how the bulk of the information is around the center
+               of the plot, and that there is little non-zero data at
+               other locations. We will window the signal to isolate the
+               bulk of the data. This limits the number of sampled values,
+               and thus reducing the number of computations.
+                 %the two parameters below will index 'original signal' around the bulk of
+                 %the information.
+                 n_window_lower = 320;
+                 n_window_upper = 2119;
+                 % window the original signal
+                 windowed_signal = original_signal(n_window_lower:n_window_upper);
+               Plot the windowed_signal and notice the difference between
+               the 'original_signal' and the 'windowed_signal'.
+                 % plot the windowed_signal as a function of n
+                 subplot(2,2,2);
+                 plot(windowed_signal);
+                 title('windowed signal');
+           b) Create a function file that will perform the DFT. A skeleton
+              of the function file is provided below. I have created the
+              function file to operate in two modes depending on how many
+              parameters you pass into the function file. All you need to
+              do is modify the code that is between the comment lines to
+              actually implement the DFT summation equation.
+                                   Comment line
+               %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+                 function [Xk, f] =  myDFT(signal,sample_rate)
+                 % Xk is an array of the DFT complex coefficients
+                 % f is an array of the frequencies that correspond to Xk
+                 % signal is a variable that contains the data points of your signal
+                 % sample_rate is the rate at which the signal was samples/created
+                 % This function file supports two modes. The first mode is the generic DFT,
+                 % and is triggered when only yhe signal is passed to the function. Ex
+                 %       myDFT(windowed_signal)
+                 % If both parameters are passed, then the function file will execute the
+                 % second mode
+                 % Ex. myDFT(windowned_signal, sample_rate);
+                 % Checks to see how many parameters were passed in. If only one parameter
+                 % is passed in, then the original DFT is calculated.
+                 if(nargin < 2)
+                     sample_rate = 1;
+                 end
+                 % The equation for the DFT is
+                 % Xk = sum(x[n]*exp(-1j*k*omega_o*n), from n = 0 to n = No-1)
+                 % No represents the number of sampled values
+                 No = length(signal);
+                 % fundamental frequency
+                 gamma_o = 2*pi/No;
+                 % n is an array that indexes the data points
+                 n = 1:length(signal);
+                 % Xn are the DFT complex coefficients
+                 Xk = zeros(1,length(signal));
+                 % f is an array that will contain the frequencies (Hz) at which the DFT is
+                 % applied
+                 f = zeros(1,length(signal));
+                 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+                 %  Student will only modify the code in between the comment lines
+                 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+                 % iterates through the harmonics of gamma_o to find the DFT complex
+                 % coefficients and the frequencies (Hz)
+                 for k = 0:No-1;
+                 Xk(k+1) = sum( %INSERT CODE HERE);
+                 f(k+1) = k*gamma_o/(2*pi)*sample_rate;
+                 end
+                 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+                 % Do not modify the code below
+                 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+                 % In the DFT, the values ranged from 0 to pi represent the positive
+                 % frequencies while the values from pi to 2pi represent the negative
+                 % frequencies Since the magnitude of the DFT is an even function we can
+                 % get all of the necessary information from looking at the values of Xn
+                 % that represent the frequencies from 0 to pi
+                 if(nargin == 2)
+                     Xk = 2*Xk(1:round(length(Xk))/2);
+                     f  = f(1:round(length(f))/2);
+                 end
+           c) Call the function to perform the DFT on your signal passing
+              in only your signal to myDFT function. Call the function
+              again, but pass in both the signal and the sample rate.
+                 % call function to perform the DFT
+                 [Xn_mode1,f_mode1] = myDFT(windowed_signal);
+                 [Xn_mode2,f_mode2] = myDFT(windowed_signal,sample_rate);
+                 % plot the DFT transform
+                 subplot(2,2,3);
+                 plot(f_mode1,abs(Xn_mode1));
+                 title('DFT mode1 complex coefficients');
+                 xlabel('frequency (hz)');
+                 ylabel('DFT complex coefficients magnitude');
+                 subplot(2,2,4);
+                 plot(f_mode2,abs(Xn_mode2));
+                 title('DFT mode 2 complex coefficients');
+                 xlabel('frequency (hz)');
+                 ylabel('DFT complex coefficients magnitude');
+           d) Question:
+) Looking at the DFT mode2 plot (where you pass in the sample
+                 rate, what frequencies are in the original signal. Hint: there are only four.
+           Remember to submit your plots and code.
+</file>
+== Signal File ==
+{{:380matlab:ch7:mt380_uy7_15_1_s_discretefouriertransform_signal.txt|}}
+== Solution Image ==
+{{:380matlab:ch7:mt380_uy7_15_1_si_discretefouriertransform_solutionimage.jpg?400|}}
+<ifauth @admin,@380ta>
+== Signal Creation ==
+{{:380matlab:solutions:ch7:mt380_uy7_15_1_sc_discretefouriertransform_signalcreation.m|}}
+== Solution ==
+DFT Function File.
+{{:380matlab:solutions:ch7:mt380_uy7_15_1_dft.m|}}
+Associated Script File
+{{:380matlab:solutions:ch7:mt380_uy7_15_1_discretefouriertransform.m|}}
+</ifauth>
+====== Extras ======
+===== Music Harmonics =====
+<file>
+   This MATLAB assignment explores the relationship between music and
+the frequency domain using the FFT. The file provided gives you a sample
+of a cello playing the note D3. An FFT of note D3 is taken in order to
+approximate Discrete Fourier Transform coefficients. These coefficients are
+used to approximate other notes pertaining to a Cello.
+    All you need to do is download the file and explore what is provided to you.
+Note, all the code is provided to you; however, you are encouraged to modify it and
+play around. Be creative.
+    As of now, you do not need to turn in anything in order to receive credit for this
+assignment.
+</file>
+== File ==
+{{:380matlab:funapplications:music_harmonics.zip|}}

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