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Discrete-Time Fourier Transform Pair

Module by: Don Johnson

Summary: Computing discrete-time frequencies by the use of Fourier transforms.

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When we obtain the discrete-time signal via sampling an analog signal, the Nyquist frequency corresponds to the discrete-time frequency 12 1 2 . To show this, note that a sinusoid at the Nyquist frequency 12 T s 1 2 T s has a sampled waveform that equals

Sinusoid at Nyquist Frequency 1/2T

cos2π12 T s n T s =cosπn=-1n 2 1 2 T s n T s n 1 n (1)

The exponential in the DTFT at frequency 12 1 2 equals ⅇ-ⅈ2πn2=ⅇ-ⅈπn=-1n 2 n 2 n 1 n , meaning that the correspondence between analog and discrete-time frequency is established:

Analog, Discrete-Time Frequency Relationship

f D = f A T s f D f A T s (2)

where f D f D and f A f A represent discrete-time and analog frequency variables, respectively. The aliasing figure provides another way of deriving this result. As the duration of each pulse in the periodic sampling signal p T s t p T s t narrows, the amplitudes of the signal's spectral repetitions, which are governed by the Fourier series coefficients of p T s t p T s t , become increasingly equal. 1 Thus, the sampled signal's spectrum becomes periodic with period 1 T s 1 T s . Thus, the Nyquist frequency 12 T s 1 2 T s corresponds to the frequency 12 1 2 .

The inverse discrete-time Fourier transform is easily derived from the following relationship:

∫-1212ⅇ-ⅈ2πfmⅇ+ⅈπfndf=1ifm=n0ifm≠n 1 2 1 2 f 2 f m f n 1 m n 0 m n (3)

Therefore, we find that

∫-1212Sⅇⅈ2πfⅇ+ⅈ2πfndf=∫-1212∑msmⅇ-ⅈ2πfmⅇ+ⅈ2πfndf=∑msm∫-1212ⅇ-ⅈ2πfm−ndf=sn f 1 2 1 2 S 2 f 2 f n f 1 2 1 2 m m s m 2 f m 2 f n m m s m f 1 2 1 2 2 f m n s n (4)

The Fourier transform pairs in discrete-time are

Fourier Transform Pairs in Discrete Time

Sⅇⅈ2πf=∑nsnⅇ-ⅈ2πfn S 2 f n n s n 2 f n (5)

Fourier Transform Pairs in Discrete Time

sn=∫-1212Sⅇⅈ2πfⅇ+ⅈ2πfndf s n f 1 2 1 2 S 2 f 2 f n (6)

Footnotes

  1. Examination of the periodic pulse signal reveals that as Δ Δ decreases, the value of c 0 c 0 , the largest Fourier coefficient, decreases to zero: | c 0 |=AΔT c 0 A Δ T . Thus, to maintain a mathematically viable Sampling Theorem, the amplitude A A must increase as 1Δ 1 Δ , becoming infinitely large as the pulse duration decreases. Practical systems use a small value of Δ Δ , say 0.1 T s 0.1 T s and use amplifiers to rescale the signal.

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