128 8. FREQUENCY DOMAIN TRANSFORMS
the frequency spectrum and allows the adjustment of the transform parameters (see Figures L8.1
through L8.3).
Figure L8.1: Android spectrogram app graphical display.
L8.1 LAB EXERCISES
1. Computational Complexity. Compare the processing time of the DFT vs. the FFT im-
plementation of the Fourier transform. Does the DFT meet real-time performance con-
straints? How does the processing time of the two implementations change as the trans-
form size is increased?
2. Linear Convolution. Consider a linear time invariant system which comprises a bandpass
filter with the following specifications:
• 8000 Hz sampling rate
• -90 dB stopband attenuation
• 800 Hz to 1800 Hz passband
• 100 Hz wide transition bands
Find the output of the system y.n/ to an input audio signal x.n/ via the overlap and add
convolution method. Record the processing time for the case with 256-sample frames as
input. It helps to use the MATLAB fdesign tool to generate the filter coefficients for this
system.
3. Frequency Domain Filtering. Solve the previous bandpass filter system in the frequency
domain by using two forward and one inverse FFT by using Y.k/ D H.k/X.k/ (convolu-
tion property). For the frequency domain case, consider the results when using 512-point
FFTs and 256-sample frames for the following two cases:
L8. LAB 8: FREQUENCY DOMAIN TRANSFORMS–DFT AND FFT 129
Figure L8.2: Android spectrogram app main screen and settings menu.
• overlap processing with 50% overlap and Hanning windowing and
• no overlap and rectangular windowing.
Use the test signals provided to examine both of the filter implementations. Once the
frequency domain filtering is completed, copy back the debugged outputs and compare
all three filtering outputs using MATLAB. Note that the frequency domain representa-
tion may be stored between calls and does not need to be re-computed after the initial
computation.
4. Reconstruction
. Using the test signals provided, how do parameter adjustments such as
the transform size, windowing function, window size, and overlap affect the spectrum?
Examine how these parameters affect the audio signal when it is synthesized from its
frequency spectrum.
130 8. FREQUENCY DOMAIN TRANSFORMS
Figure L8.3: iOS spectrogram app graphical display.
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