Discrete-Time Fourier Transform (DTFT)

The Fourier transform of the discrete-time signal

is defined to be

Frequency here has no units. As should be expected, this definition is linear, with the transform of a sum of signals equaling the sum of their transforms. Real-valued signals have conjugate-symmetric spectra:

Exercise

A special property of the discrete-time Fourier transform is that it is periodic with period one:

. Derive this property from the definition of the DTFT.

Because of this periodicity, we need only plot the spectrum over one period to understand completely the spectrum's structure; typically, we plot the spectrum over the frequency range

. When the signal is real-valued, we can further simplify our plotting chores by showing the spectrum only over

; the spectrum at negative frequencies can be derived from positive-frequency spectral values.

When we obtain the discrete-time signal via sampling an analog signal, the Nyquist frequency corresponds to the discrete-time frequency

. To show this, note that a sinusoid having a frequency equal to the Nyquist frequency has a sampled waveform that equals

The exponential in the DTFT at frequency

equals

, meaning that discrete-time frequency equals analog frequency multiplied by the sampling interval

and

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

narrows, the amplitudes of the signal's spectral repetitions, which are governed by the Fourier series coefficients of

, become increasingly equal. Examination of the periodic pulse signal reveals that as

decreases, the value of

, the largest Fourier coefficient, decreases to zero:

. Thus, to maintain a mathematically viable Sampling Theorem, the amplitude

must increase as

, becoming infinitely large as the pulse duration decreases. Practical systems use a small value of

say and use amplifiers to rescale the signal. Thus, the sampled signal's spectrum becomes periodic with period

. Thus, the Nyquist frequency

corresponds to the frequency

Let's compute the discrete-time Fourier transform of the exponentially decaying sequence

, where

is the unit-step sequence. Simply plugging the signal's expression into the Fourier transform formula,

This sum is a special case of the geometric series.

Thus, as long as

, we have our Fourier transform.

Using Euler's relation, we can express the magnitude and phase of this spectrum.

No matter what value of

we choose, the above formulae clearly demonstrate the periodic nature of the spectra of discrete-time signals. Figure shows indeed that the spectrum is a periodic function. We need only consider the spectrum between

and

to unambiguously define it. When

, we have a lowpass spectrum—the spectrum diminishes as frequency increases from 0 to

—with increasing

leading to a greater low frequency content; for

, we have a highpass spectrum (Figure 2).

Spectrum of exponential signal

Figure 1. The spectrum of the exponential signal (

) is shown over the frequency range [-2, 2], clearly demonstrating the periodicity of all discrete-time spectra. The angle has units of degrees.

Spectra of exponential signals

Figure 2. The spectra of several exponential signals are shown. What is the apparent relationship between the spectra for

and

Exercise

Analogous to the analog pulse signal, let's find the spectrum of the length-

pulse sequence.

The Fourier transform of this sequence has the form of a truncated geometric series.

For the so-called finite geometric series, we know that

for all values of α.

Derive this formula for the finite geometric series sum. The "trick" is to consider the difference between the series' sum and the sum of the series multiplied by

which, after manipulation, yields the geometric sum formula.

Applying this result yields (Figure 3).

The ratio of sine functions has the generic form of

, which is known as the discrete-time sinc function

. Thus, our transform can be concisely expressed as

. The discrete-time pulse's spectrum contains many ripples, the number of which increase with

, the pulse's duration.

Spectrum of length-ten pulse

Figure 3. The spectrum of a length-ten pulse is shown. Can you explain the rather complicated appearance of the phase?

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

Therefore, we find that

The Fourier transform pairs in discrete-time are

The properties of the discrete-time Fourier transform mirror those of the analog Fourier transform. The DTFT properties table shows similarities and differences. One important common property is Parseval's Theorem.

To show this important property, we simply substitute the Fourier transform expression into the frequency-domain expression for power.

Using the orthogonality relation, the integral equals

, where

is the unit sample. Thus, the double sum collapses into a single sum because nonzero values occur only when

, giving Parseval's Theorem as a result. We term

the energy in the discrete-time signal

in spite of the fact that discrete-time signals don't consume (or produce for that matter) energy. This terminology is a carry-over from the analog world.

Exercise

Suppose we obtained our discrete-time signal from values of the product

, where the duration of the component pulses in

. How is the discrete-time signal energy related to the total energy contained in

? Assume the signal is bandlimited and that the sampling rate was chosen appropriate to the Sampling Theorem's conditions.

If the sampling frequency exceeds the Nyquist frequency, the spectrum of the samples equals the analog spectrum, but over the normalized analog frequency