Now that we know what the DFT frequency response is to a complex sinusoidal input, it's easy to determine the DFT frequency response to a real input sequence. Say we want the DFT's response to a real cosine sequence, like that shown in Figure 3-40(a), expressed as

where k is the integral number of complete cycles occurring in the N samples. Remembering Euler's relationship cos(ø) = (ejø + e–jø)/2, we can show the desired DFT as Xr(m) where

Equation 3-72

Fortunately, in the previous section we just finished determining the closed form of a summation expression like those in Eq. (3-72), so we can write the closed form for Xr(m) as

Equation 3-73

We show the magnitude of those two ratio of sines terms as the sinc functions in Figure 3-43. Here again, the DFT is sampling the input cosine sequence's continuous spectrum and, because k = m, only one DFT bin is nonzero. Because the DFT's input sequence is real, Xr(m) has both positive and negative frequency components. The positive frequency portion of Figure 3-43 corresponds to the first ratio of sines term in Eq. (3-73) and the second ratio of sines term in Eq. (3-73) produces the negative frequency components of Xr(m).

Figure 3-43. N-point DFT frequency magnitude response to a real cosine having integral k cycles in the N-point time sequence xr(n) = cos(2pnk/N).



DFT leakage is again demonstrated if our input sequence frequency were shifted from the center of the kth bin to k+0.25 as shown in Figure 3-44. (We used this concept of real input DFT amplitude response to introduce the effects of DFT leakage in Section 3.8.)

Figure 3-44. N-point DFT frequency magnitude response showing spectral leakage of a real cosine having k+0.25 cycles in the N-point time sequence xr(n).

In Table 3-4, the various mathematical expressions for the (positive frequency) amplitude response of the DFT to a real cosine input sequence are simply those expressions in Table 3-3 reduced in amplitude by a factor of 2.

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Chapter One. Discrete Sequences and Systems

Chapter Two. Periodic Sampling

Chapter Three. The Discrete Fourier Transform

Chapter Four. The Fast Fourier Transform

Chapter Five. Finite Impulse Response Filters

Chapter Six. Infinite Impulse Response Filters

Chapter Seven. Specialized Lowpass FIR Filters

Chapter Eight. Quadrature Signals

Chapter Nine. The Discrete Hilbert Transform

Chapter Ten. Sample Rate Conversion

Chapter Eleven. Signal Averaging

Chapter Twelve. Digital Data Formats and Their Effects

Chapter Thirteen. Digital Signal Processing Tricks

Appendix A. The Arithmetic of Complex Numbers

Appendix B. Closed Form of a Geometric Series

Appendix C. Time Reversal and the DFT

Appendix D. Mean, Variance, and Standard Deviation

Appendix E. Decibels (dB and dBm)

Appendix F. Digital Filter Terminology

Appendix G. Frequency Sampling Filter Derivations

Appendix H. Frequency Sampling Filter Design Tables

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Understanding Digital Signal Processing
Understanding Digital Signal Processing (2nd Edition)
ISBN: 0131089897
EAN: 2147483647
Year: 2004
Pages: 183
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