EE313 Linear Systems & Signals - Mini-Project #1 Hints

• Scope and format. Please see the homework hints page or the last two pages of the Mini-Project #1 assignment.

• Finding a harmonic frequency in the Fourier series coefficient vector. In continuous time, the duration between samples is the sampling time, T_s. Since there are N = 38500 samples in the violin sound, the violin sound lasts for N T_s or 3.4921 seconds.

  The Fourier series assumes the violin sound is periodic with periodicity N T_s seconds. Hence, the fundamental period is T₀ = N T_s seconds and the fundamental frequency is

  f₀ = 1 / T₀ = 1 / (N T_s) = f_s / N

  The FFT takes the vector of N samples for the recorded signal, and returns the N exponential Fourier series coefficients.

  fourierSeriesCoeffs = fft(instrumentSound);
  

  In the discrete-time case, we have a finite number of exponential Fourier series coefficients unlike the continuous-time Fourier series which has infinite terms.

  The first half of the fourierSeriesCoeffs vector contains the Fourier series coefficients for non-negative frequencies, and the second half of the vector contains the Fourier series coefficients for negative frequencies. In Matlab, the first element is at index 1. In fourierSeriesCoeffs,

  • index #1 contains a₀ which is the Fourier series coefficient for frequency 0 Hz. This is the average value (mean) in the violin sound.
  • index #2 contains a₁ which is the Fourier series coefficient for frequency (f_s / N)
  • index #3 contains a₂ which is the Fourier series coefficient for frequency 2 (f_s / N)
  • ...
  • index #N/2 contains a_(N/2)-1 which is the Fourier series coefficient for frequency (N/2-1) (f_s / N) = (1/2) f_s - f_s / N
  • index #N/2 + 1 contains a_N/2 which is the Fourier series coefficient for frequency -N/2 (f_s / N) = -(1/2) f_s.
  • ...
  • index #N contains a_-1 which is the Fourier series coefficient for frequency -f_s / N

  The kth harmonic frequency is f_k = k f₀ where f₀ = f_s / N.

  If the frequency is non-negative, then we can convert nonNegativeFrequency to an FFT vector index into by

  N = length(instrumentSound);
  f0 = fs / N;
  fftIndex = round(nonNegativeFrequency / f0) + 1;
  

  and compute

  fftCoeffGain = abs(fourierSeriesCoeffs(fftIndex));
  fftCoeffPhase = angle(fourierSeriesCoeffs(fftIndex));
  

  Please see Appendix A in the mini-project #1 assignment for more information about using the FFT to compute the Fourier series.

• Section 2.2 Time-Domain Analysis. The MATLAB code introduces a variable N for the number of samples in the recorded speech. This will clash with the N used in the sum of sinusoids formula that appears in Sections 1 and 3. They are different quantities.

• Section 2.3 Frequency-Domain Analysis. Part (a) plots the absolute value of the exponential Fourier series coefficients of the violin sound computed using the Fast Fourier Transform (FFT). When plotting the Fourier series coefficients computed by the FFT, we use the FFT shift command fftshift to swap the halves of the fourierSeriesCoeffs vector so negative, zero, and positive frequencies are plotted.

  In part (b), you'll use the Data Tips tool to find the horizontal coordinate (frequency) and vertical coordinate (absolute value of the Fourier series coefficient) from the frequency domain plot. The absolute value is the gain. To find the phase, you'll need to convert the frequency to an index into the fourierSeriesCoeffs vecor using the information in the section above "Finding a harmonic frequency in the Fourier series coefficient vector".

  In part (d), you're asked to find "for what range of time is the principal frequency present". The principal frequency is 261.63 Hz. You can zoom into the spectrogram to see over what time the principal frequency is present.

• 3.0 Synthesizing Sound. In part (a), you'll manually select the frequencies at the strongest peaks and use the section "Finding a harmonic frequency in the Fourier series coefficient vector" to find the Fourier series coefficient corresponding to that frequency.

  In part (c), you'll need to determine the range of FFT indices that corresponds to a range of frequencies from -7.5 Hz to 7.5 Hz.

  During playback of the synthesized signal, please use the soundsc command instead of the sound command.

• Office Hour Notes. Marker board notes.

bevans@ece.utexas.edu