0.1 Discrete-time signals

Although the discrete-time signal $x\left(n\right)$ could be any ordered sequence of numbers, they are usually samples of a continuous-timesignal. In this case, the real or imaginary valued mathematical function $x\left(n\right)$ of the integer $n$ is not used as an analogy of a physical signal, but as some representation of it (such as samples).In some cases, the term digital signal is used interchangeably with discrete-time signal, or the label digitalsignal may be use if the function is not real valued but takes values consistent with some hardware system.

Indeed, our very use of the term “discrete-time" indicates the probable origin of the signals when, in fact, the independentvariable could be length or any other variable or simply an ordering index. The term “digital" indicates the signal is probably going tobe created, processed, or stored using digital hardware. As in the continuous-time case, the Fourier transform will again be ourprimary tool [link] , [link] , [link] .

Notation has been an important element in mathematics. In some cases, discrete-time signals are best denoted as a sequence ofvalues, in other cases, a vector is created with elements which are the sequence values. In still other cases, a polynomial is formedwith the sequence values as coefficients for a complex variable. The vector formulation allows the use of linear algebra and thepolynomial formulation allows the use of complex variable theory.

The discrete fourier transform

The description of signals in terms of their sinusoidal frequency content has proven to be as powerful and informative for discrete-time signals asit has for continuous-time signals. It is also probably the most powerful computational tool we will use. We now develop the basicdiscrete-time methods starting with the discrete Fourier transform (DFT) applied to finite length signals, followed by the discrete-time Fouriertransform (DTFT) for infinitely long signals, and ending with the z-transform which uses the powerful tools of complex variable theory.

Definition of the dft

It is assumed that the signal $x\left(n\right)$ to be analyzed is a sequence of $N$ real or complex values which are a function of the integer variable $n$ . The DFT of $x\left(n\right)$ , also called the spectrum of $x\left(n\right)$ , is a length $N$ sequence of complex numbers denoted $C\left(k\right)$ and defined by

using the usual engineering notation: $j=\sqrt{-1}$ . The inverse transform (IDFT) which retrieves $x\left(n\right)$ from $C\left(k\right)$ is given by

which is easily verified by substitution into [link] . Indeed, this verification will require using the orthogonality of the basis function ofthe DFT which is

The exponential basis functions, $e^{-j\frac{2\pi }{N}k}$ , for $k\in \{0,N-1\}$ , are the $N$ values of the $N$ th roots of unity (the N zeros of the polynomial ${(s-1)}^N$ ). This property is what connects the DFT to convolution and allows efficient algorithmsfor calculation to be developed [link] . They are used so often that the following notation is defined by

with the subscript being omitted if the sequence length is obvious from context. Using this notation, the DFT becomes