We have talked about the concept of frequency and sampling a continuous-time signal to generate a discrete-time signal. Remember that the reason we work with discrete-time signals is that finite computer memory can store only a fixed number of time values. Similarly, this finite memory can also store only a fixed number of frequency values instead of infinite range of .

For this reason, while we are at sampling in time domain, we also want to sample the frequency domain. Assume that a total of samples were collected in time domain (see this Figure) at a rate of , thus spanning a time duration of seconds. Then, the lowest frequency that can be represented by these samples is the one by a signal that completes one full cycle — and no more — during this interval of seconds and consequently given by Hz. Consider the equation

and observe that

- Frequency resolution, determined by the lowest frequency that can be represented, is given by .
- Viewed as , we can get discrete frequency samples that are integer multiples of , if we want the same number of frequency samples as the time domain samples.

As explained in the article on sampling, the unique range of continuous frequency is , or

called the primary zone. To obtain the discrete frequency axis, is divided into equal intervals (assuming is even) by sampling at instants

(1)

resulting in a discrete frequency resolution of . So discrete frequency is basically the spectral content in the primary zone sampled at equally spaced frequency points. The term is absent because it is the same as owing to the axis periodicity.

Eq (1) can also be written as

(2)

Consequently, the index of discrete frequency axis shown in Figure below is given by , or

(3)

Comparing Eq (1) and Eq (2), we arrive at the relation between discrete frequency and continuous frequency as

From the above equation, the actual continuous frequency represented by discrete frequency index depends on both the sample rate and as

For example, with kHz and ,

Each is therefore called a frequency bin and the value determines the number of input samples and the resolution of discrete frequency domain. Understanding the sampling theorem and the relationship between continuous and discrete frequencies will make the further concepts much easier to grasp.

The units of discrete frequency from Eq (4) are

Eq (4) is one of the two most fundamental relations in digital signal processing, the other being the sampling theorem. These are the two interfaces between continuous and discrete worlds.

Establishing the relationship between continuous and discrete frequencies gives the frequency domain as a sequence of numbers stored in a processor memory.

When in doubt about continuous and discrete frequency domains, refer to Eq (4)!

Having known the discrete frequency axis, it is natural to ask how these discrete frequencies physically look like. Of course, these are frequencies of a set of complex sinusoids with both and components, as defined in a previous post about complex sinusoids and shown in Figure below for .

Following are a few key observations in this figure.

- For , the sinusoid is because , while sinusoid is because . Of course in a time -plane, this means a point standing still at .
- The and waveforms for are and and they complete on full cycle during samples. This is true in general. Here for a sample rate of kHz, it resulted in kHz with a time period equal to ms. One full cycle ensues because samples at kHz span a time duration of ms as well. Similarly, each complex sinusoid for each span complete cycles in an interval of samples.
- For negative values of , sinusoids remain unchanged while sinusoids change sign. This can be observed from the definition of a complex sinusoid rotating in an -plane as shown in this Figure and this Figure. part is same for both positive and negative directions of rotations, whereas part is positive for positive direction and negative for negative direction.

In conclusion, the range of unique discrete frequencies is

(5)

From the above discussion, the rate of oscillation in discrete-time signals decreases with going from to and increases from to , remembering that is the same as .

Consider a signal

It is clear that continuous frequencies present in this signal are kHz, kHz and kHz. Since the maximum frequency is kHz, the sampling theorem gives the Nyquist rate as kHz = kHz.

Now suppose that this signal is sampled at kHz, the folding frequency is then given by kHz. Sampling the signal at equal intervals of

If this signal is reconstructed in continuous-time domain, the frequencies present are and . From Eq (4), these appear as continuous frequencies at kHz and kHz. Since only kHz is less than and hence within aliasing-free range, it is still there after sampling. The other two are above the folding frequency and hence aliased to kHz and kHz.

In these articles, every time we write the expression involving a discrete frequency, we express it either as or , instead of that is used in most texts. The former clearly indicates the presence of a sampled discrete frequency analogous to the variable in , while it is easy to lose this meaning in the latter (the expression more closely resembles a Fourier series description, which we keep out to keep the explanations simple). In addition, dealing with complex numbers through an notation not only describes how they are implemented in actual electronic circuits, but it also constantly reminds us how phase is equally important in signal analysis. It is easy to overlook this fact in other more commonly used notations.