Depending on whether a signal is periodic and discrete, its Fourier transform will take one of the four forms:

**I. Aperiodic, continuous time signal , continuous, aperiodic spectrum**This is the most general form of Fourier transform:

**II. Aperiodic, discrete time signal , continuous, periodic spectrum**The discrete time signal can be considered as a sequence of samples of a continuous signal. The time interval between two consecutive samples and is , where is the sampling rate, which is also the period of the spectrum in the frequency domain.

The discrete time signal can be written as

and its transform is:

**III. Periodic, continuous time signal , discrete, aperiodic spectrum**This is the Fourier series expansion of periodic signals. The time period is , and the interval between two consecutive frequency components is , and its transform is:

The discrete spectrum can also be represented as:

**IV. Periodic, discrete time signal , discrete, periodic spectrum**

where is the number of samples in each period in both temporal and frequency domains. Since , the above can be rewritten as:

The coefficients for forward and inverse transforms can be either and 1, or for both of them to make the transform pair perfectly symmetric. These scaling factors are not essential in practice.

**Four forms of Fourier transform** (Table 5.3):

**The Convolution theorem**

**Parseval's formula**