Discrete Fourier Transform and Signal Spectrum - Digital Signal Processing: Fundamentals and Applications - page 135

obtained using a sampling rate of f s ¼ 8 ; 000 Hz, we apply the DFT to compute the
amplitude spectrum.
a. Determine the frequency resolution when the data length is 100 samples. Without using
the window function, is there any spectral leakage in the computed spectrum?
Explain.
b. Determine the frequency resolution when the data length is 73 samples. Without using the
window function, is there any spectral leakage in the computed spectrum?
Explain.
4.21. Given a sequence xðnÞ for 0 n 3, where xð 0 Þ¼ 4, xð 1 Þ¼ 3, xð 2 Þ¼ 2, and xð 3 Þ¼ 1,
evaluate its DFT XðkÞ using the decimation-in-frequency FFT method, and determine the
number of complex multiplications.
4.22. Given the DFT sequence XðkÞ for 0 k 3 obtained in Problem 4.21, evaluate its inverse
DFT xðnÞ using the decimation-in-frequency FFT method.
4.23. Given a sequence xðnÞ for 0 n 3, where xð 0 Þ¼ 0 : 8, xð 1 Þ¼ 0 : 6, xð 2 Þ¼ 0 : 4, and
xð 3 Þ¼ 0 : 2, evaluate its DFT XðkÞ using the decimation-in-frequency FFT method, and
determine the number of complex multiplications.
4.24. Given the DFT sequence XðkÞ for 0 k 3 obtained in Problem 4.23, evaluate its inverse
DFT xðnÞ using the decimation-in-frequency FFT method.
4.25. Given a sequence xðnÞ for 0 n 3, where xð 0 Þ¼ 4, xð 1 Þ¼ 3, xð 2 Þ¼ 2, and xð 3 Þ¼ 1,
evaluate its DFT XðkÞ using the decimation-in-time FFT method, and determine the number
of complex multiplications.
4.26. Given the DFT sequence XðkÞ for 0 k 3 computed in Problem 4.25, evaluate its inverse
DFT xðnÞ using the decimation-in-time FFT method.
4.27. Given a sequence xðnÞ for 0 n 3, where xð 0 Þ¼ 0 : 8, xð 1 Þ¼ 0 : 4, xð 2 Þ¼ 0 : 4, and
xð 3 Þ¼ 0 : 2, evaluate its DFT XðkÞ using the decimation-in-time FFT method, and
determine the number of complex multiplications.
4.28. Given the DFT sequence XðkÞ for 0 k 3 computed in Problem 4.27, evaluate its inverse
DFT xðnÞ using the decimation-in-time FFT method.
4.7.1 Computer Problems with MATLAB
Use MATLAB to solve Problems 4.29 and 4.30.
4.29. Consider three sinusoids with the following amplitudes and phases:
x 1 ðtÞ¼ 5cos ð 2 pð 500 ÞtÞ
x 2 ðtÞ¼ 5cos ð 2 pð 1200 Þt þ 0 : 25 pÞ
x 3 ðtÞ¼ 5cos ð 2 pð 1800 Þt þ 0 : 5 pÞ
a. Create a MATLAB program to sample each sinusoid and generate a sum of three sinu-
soids, that is, xðnÞ¼x 1 ðnÞþx 2 ðnÞþx 3 ðnÞ , using a sampling rate of 8,000 Hz. Plot xðnÞ
over a range of 0.1 seconds.
b. Use the MATLAB function fft() to compute DFT coefficients, and plot and examine the
spectrum of the signal xðnÞ .