Complex Digital Signal Processing in Telecommunications - Applications of Digital Signal Processing - page 9

1.3 Complex digital processing techniques - complex Fourier transforms
Digital systems and signals can be represented in three domains - time domain, z-domain
and frequency domain. To cross from one domain to another, the Fourier and z-transforms
are employed (Fig. 5). Both transforms are fundamental building-blocks of signal processing
theory and exist in two formats - forward and inverse (Smith, 1999).

Frequency
Domain

Fourier
transforms

Time
Domain

Z-
transforms

Z-
Domain

Fig. 5. Relationships between frequency, time, and z- domains

The Fourier transforms group contains four families, which differ from one another in the
type of time-domain signal which they process - periodic or aperiodic and discrete or
continuous . Discrete Fourier Transform (DFT) deals with discrete periodic signals, Discrete
Time Fourier Transform (DTFT) with discrete aperiodic signals, and Fourier Series and
Fourier Transform with periodic and aperiodic continuous signals respectively. In addition to
having forward and inverse versions, each of these four Fourier families exists in two
forms - real and complex , depending on whether real or complex number math is used. All
four Fourier transform families decompose signals into sine and cosine waves; when these
are expressed by complex number equations, using Euler's identity, the complex versions of
the Fourier transforms are introduced.
DFT is the most often-used Fourier transform in DSP. The DFT family is a basic
mathematical tool in various processing techniques performed in the frequency domain, for
instance frequency analysis of digital systems and spectral representation of discrete signals.
In this chapter, the focus is on complex DFT. This is more sophisticated and wide-ranging
than real DFT, but is based on the more complicated complex number math. However,
numerous digital signal processing techniques, such as convolution, modulation,
compression, aliasing, etc. can be better described and appreciated via this extended math.
(Sklar, 2001)
Complex DFT equations are shown in Table 1. The forward complex DFT equation is also
called analysis equation. This calculates the frequency domain values of the discrete periodic
signal, whereas the inverse ( synthesis ) equation computes the values in the time domain.

Table 1. Complex DFT transforms in rectangular form

The time domain signal x ( n ) is a complex discrete periodic signal; only an N -point unique
discrete sequence from this signal, situated in a single time-interval (0÷ N , - N /2÷ N /2, etc.) is