7: Discrete -Time Fourier Transform (DTFT)

The Fourier transform of the discrete-time signal s(n) is defined to be

\[S(e^{i2\pi f})=\sum_{n=-\infty }^{\infty }s(n)e^{-(i2\pi fn)} \nonumber \]

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:

\[S(e^{-(i2\pi f)})=\overline{S(e^{j2\pi f}}) \nonumber \]

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

\[\left [ -\frac{1}{2},\frac{1}{2} \right ] \nonumber \]

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

\[\left [ 0,\frac{1}{2} \right ] \nonumber \]

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 1/2. To show this, note that a sinusoid having a frequency equal to the Nyquist frequency 1/2T_s has a sampled waveform that equals

\[\cos \left ( 2\pi \times \frac{1}{2T_{s}}nT_{s} \right )=\cos (\pi n)=(-1)^{n} \nonumber \]

The exponential in the DTFT at frequency 1/2 equals

\[e^{-\frac{i2\pi n}{2}}=e^{-(i\pi n)}=(-1)^{n} \nonumber \]

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

\[f_{D}=f_{A}T_{s} \nonumber \]

f_D and 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_Ts(t) narrows, the amplitudes of the signal's spectral repetitions, which are governed by the Fourier series coefficients of p_Ts(t), become increasingly equal. Examination of the periodic pulse signal reveals that as Δ decreases, the value of c₀, the largest Fourier coefficient, decreases to zero:

\[\left | c_{0} \right |=\frac{A\Delta }{T_{s}} \nonumber \]

Thus, to maintain a mathematically viable Sampling Theorem, the amplitude A must increase as 1/Δ, becoming infinitely large as the pulse duration decreases. Practical systems use a small value of Δ, say 0.1.T_s and use amplifiers to rescale the signal. Thus, the sampled signal's spectrum becomes periodic with period 1/T_s. Thus, the Nyquist frequency 1/2T_s corresponds to the frequency ½.

Applying this result yields to Figure 5.6.3,

\[S(e^{i2\pi f})=\frac{1-e^{-(i2\pi fN)}}{1-e^{-(i2\pi f)}} \nonumber \]

\[S(e^{i2\pi f})=e^{-(i\pi f(N-1))}\frac{\sin (\pi fN)}{\sin (\pi f)} \nonumber \]

The ratio of sine functions has the generic form of

\[\frac{\sin (Nx)}{\sin (x)} \nonumber \]

which is known as the discrete-time sinc function dsinc (x). Thus, our transform can be concisely expressed as

\[S(e^{i2\pi f})=e^{-(i\pi f(N-1))}dsinc(\pi f) \nonumber \]

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

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

\[\int_{-\frac{1}{2}}^{\frac{1}{2}} e^{-(i2\pi fm)} e^{i2\pi fn}df=\begin{cases} 1 & \text{ if } m=n \\ 0 & \text{ if } m\neq n \end{cases} \nonumber \]

\[\int_{-\frac{1}{2}}^{\frac{1}{2}} e^{-(i2\pi fm)} e^{i2\pi fn}df=\delta (m-n) \nonumber \]

Therefore, we find that

\[\int_{-\frac{1}{2}}^{\frac{1}{2}} S(e^{i2\pi f})e^{i2\pi fn}df=\int_{-\frac{1}{2}}^{\frac{1}{2}} \sum_{_{m}m}s(m)e^{-(i2\pi fm)}e^{i2\pi fn}df \nonumber \]

\[\int_{-\frac{1}{2}}^{\frac{1}{2}} S(e^{i2\pi f})e^{i2\pi fn}df=\sum_{_{m}m}s(m)\int_{-\frac{1}{2}}^{\frac{1}{2}} e^{(-(i2\pi f))(m-n)}df \nonumber \]

\[\int_{-\frac{1}{2}}^{\frac{1}{2}} S(e^{i2\pi f})e^{i2\pi fn}df=s(n) \nonumber \]

The Fourier transform pairs in discrete-time are

\[S(e^{i2\pi f})=\sum_{n=-\infty }^{\infty }s(n)e^{-(i2\pi fn)} \nonumber \]

\[s(n)=\int_{-\frac{1}{2}}^{\frac{1}{2}} S(e^{i2\pi f})e^{i2\pi fn}df \nonumber \]

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

\[\sum_{n=-\infty }^{\infty }\left ( \left | s(n) \right | \right )^{2}=\int_{-\frac{1}{2}}^{\frac{1}{2}}\left | S(e^{i2\pi f}) \right |^{2}df \nonumber \]

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

\[\int_{-\frac{1}{2}}^{\frac{1}{2}}\left | S(e^{i2\pi f}) \right |^{2}df=\int_{-\frac{1}{2}}^{\frac{1}{2}}\sum_{nn}s(n)e^{-(i2\pi fn)}\sum_{mm}\overline{s(n)}e^{i2\pi fm}df \nonumber \]

\[\int_{-\frac{1}{2}}^{\frac{1}{2}}\left | S(e^{i2\pi f}) \right |^{2}df=\sum_{n,m,n,m}s(n)\overline{s(n)}\int_{-\frac{1}{2}}^{\frac{1}{2}}e^{i2\pi f(m-n)}df \nonumber \]

Using the orthogonality relation, the integral equals δ(m-n), where δ(n) is the unit sample. Thus, the double sum collapses into a single sum because nonzero values occur only when n=m, giving Parseval's Theorem as a result. We term

\[\sum_{nn}s^{2}(n) \nonumber \]

the energy in the discrete-time signal s(n) 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.