1.1 Discrete-time signals  (Page 5/10)

Definition of the dtft

The DTFT of a possibly infinitely long real (or complex) valued sequence $f\left(n\right)$ is defined to be

and its inverse denoted IDTFT is given by

Verification by substitution is more difficult than for the DFT. Here convergence and the interchange of order of the sum andintegral are serious questions and have been the topics of research over many years. Discussions of the Fourier transform and series forengineering applications can be found in [link] , [link] . It is necessary to allow distributions or delta functions to be used to gain the fullbenefit of the Fourier transform.

Note that the definition of the DTFT and IDTFT are the same as the definition of the IFS and FS respectively. Since the DTFT is a continuousperiodic function of $\omega$ , its Fourier series is a discrete set of values which turn out to be the original signal. This duality can behelpful in developing properties and gaining insight into various problems. The conditions on a function to determine if it can be expandedin a FS are exactly the conditions on a desired frequency response or spectrum that will determine if a signal exists to realize or approximateit.

Properties

The properties of the DTFT are similar to those for the DFT and are important in the analysis and interpretation of long signals. The mainproperties are given here using the notation that the DTFT of a complex sequence $x\left(n\right)$ is $\mathcal{F}\left\{x\right(n\left)\right\}=X\left(\omega \right)$ .

1. Linear Operator: $\mathcal{F}\{x+y\}=\mathcal{F}\left\{x\right\}+\mathcal{F}\left\{y\right\}$
2. Periodic Spectrum: $X\left(\omega \right)=X(\omega +2\pi )$
3. Properties of Even and Odd Parts: $x\left(n\right)=u\left(n\right)+jv\left(n\right)$ and $X\left(\omega \right)=A\left(\omega \right)+jB\left(\omega \right)$

   $u$	$v$	$A$	$B$	$\left|X\right|$	$\theta$
   even	0	even	0	even	0
   odd	0	0	odd	even	0
   0	even	0	even	even	$\pi /2$
   0	odd	odd	0	even	$\pi /2$

4. Convolution: If non-cyclic or linear convolution is defined by:
   $y\left(n\right)=h\left(n\right)*x\left(n\right)={\sum }_{m=-\infty }^{\infty }h(n-m)x\left(m\right)={\sum }_{k=-\infty }^{\infty }h\left(k\right)x(n-k)$
   then $\mathcal{F}\left\{h\right(n)*x(n\left)\right\}=\mathcal{F}\left\{h\right(n\left)\right\}\mathcal{F}\left\{x\right(n\left)\right\}$
5. Multiplication: If cyclic convolution is defined by:
   $Y\left(\omega \right)=H\left(\omega \right)\circ X\left(\omega \right)={\int }_0^T\tilde{H}(\omega -\Omega )\tilde{X}\left(\Omega \right)d\Omega$
   $\mathcal{F}\left\{h\left(n\right)x\left(n\right)\right\}=\frac{1}{2\pi }\mathcal{F}\left\{h\left(n\right)\right\}\circ \mathcal{F}\left\{x\left(n\right)\right\}$
6. Parseval: ${\sum }_{n=-\infty }^{\infty }{\left|x\left(n\right)\right|}^2=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }{\left|X\left(\omega \right)\right|}^{2}d\omega$
7. Shift: $\mathcal{F}\left\{x(n-M)\right\}=X\left(\omega \right)e^{-j\omega M}$
8. Modulate: $\mathcal{F}\left\{x\left(n\right)e^{j{\omega }_{0}n}\right\}=X(\omega -{\omega }_0)$
9. Sample: $\mathcal{F}\left\{x\left(Kn\right)\right\}=\frac{1}{K}{\sum }_{m=0}^{K-1}X(\omega +Lm)$ where $N=LK$
10. Stretch: $\mathcal{F}\left\{x_s\left(n\right)\right\}=X\left(\omega \right)$ , for $-K\pi \le \omega \le K\pi$ where $x_s\left(Kn\right)=x\left(n\right)$ for integer $n$ and zero otherwise.
11. Orthogonality: ${\sum }_{n=-\infty }^{\infty }{e}^{-j{\omega }_{1}n}{e}^{-j{\omega }_{2}n}=2\pi \delta ({\omega }_1-{\omega }_2)$

Evaluation of the dtft by the dft

If the DTFT of a finite sequence is taken, the result is a continuous function of $\omega$ . If the DFT of the same sequence is taken, the results are $N$ evenly spaced samples of the DTFT. In other words, the DTFT of a finite signal can be evaluated at $N$ points with the DFT.

and because of the finite length

If we evaluate $\omega$ at $N$ equally space points, this becomes

which is the DFT of $x\left(n\right)$ . By adding zeros to the end of $x\left(n\right)$ and taking a longer DFT, any density of points can be evaluated. This isuseful in interpolation and in plotting the spectrum of a finite length signal. This is discussed further in Sampling, Up-Sampling, Down-Sampling, and Multi-Rate Processing .

There is an interesting variation of the Parseval's theorem for the DTFT of a finite length- $N$ signal. If $x\left(n\right)\ne 0$ for $0\ge n\ge N-1$ , and if $L\ge N$ , then