Digital Signal Processing: An Introductory Note

Digital Signal Processing (DSP) comprises the techniques and algorithms for transforming, filtering and representing digital signals (DSP is a subfield of the more general Signal Processing topic).

Continuous and Digital Signal Processing

A signal is a measurement of a process, an observation of the behavior of some system. Numerically, a signal is a time-varying or spatial-varying quantity (in the following, for simplicity, we’ll assume that the independent variable is time t). Some physical signals, such as speech and image are continuous in time. For instance, the speech signal is a continuously varying acoustic pressure wave. Sometimes, continuous signals x(t) are referred to as continuous or analog waveforms (continuous and analog are typically interchangeable terms albeit analog is a kind of absolute term, and we will not be using it in the following). Continuous signals vary at an uncountable infinite number of times. On its side, digital processing units can only handle sequences of numbers, i.e., they are discrete devices. In order to harness the benefits of digital processing units, continuous signals have to be first discretized (sampled). After sampling, we get a digital signal, which we might use as a representation of the original continuous signal. This sampling process is performed by a Continuous-to-Discrete (C/D) converter.

Summarizing, sampling is the process by which a digital representation of a continuous time signal is obtained. Basically, during sampling we select a finite number of data points (in a finite time interval) to represent the infinite amount of data that the continuous signal contains (within the same interval). If sampling is periodic, we sample x(t) at uniformly spaced time instants. Sampling is by no means a trivial issue, and we have to be careful in selecting the discrete data values… how well does this discrete sequence represent the continuous signal?.

This is other important theoretical subtlety of sampling. The C/D sampler that produces the discrete signal is (theoretically) able to handle infinite amplitude precision. Thereby, although the digital signal x[n] is discrete in time, it is continuous in amplitude. This type of signal is called discrete-time signal. Nevertheless, this C/D is an ideal device. In practice, we recur to an Analog-To-Digital (A/D) converter which quantizes each amplitude to a finite set of values closest to the actual continuous signal amplitude. This is the digital signal, which has both the independent time variable and its amplitude quantized. In practice, we work mostly with digital signals.

The Continuous Sinewave

One of the most simple yet useful signal is the sinewave, which is a member of the family of functions expressed as:

$latex x(t) = A\sin(2 \pi f_0 t)$

A is the amplitude, and we can discard it for the time being. Needless to say, theses are periodic functions whose frequency is given by $latex f_0$ (in Hz). For instance, if $latex f_0 = 2 Hz$, our sinewave would complete two oscillations in 1 second. The duration of an oscillation (the period) is the reciprocal of its frequency, $latex 1/f0$.

Now, in order to derive a digital representation of such sinewave, we will take regularly spaced samples. The time elapsed between samples is given by the parameter $latex t_s$ (the sampling period) whose reciprocal, $latex 1/t_s$, is the sampling frequency $latex f_s$. Then, our discrete signal is:

$latex x[n] = A \sin (2 \pi f_0 n t_s)$

x[n] is just a sequence of numbers. The first number of such sequence is x[0], the second is x[1], and so further. Therefore, n takes on only integer values 0, 1, 2, … Notice that similarly to $latex 2 \pi f_0 t$, $latex 2 \pi f_0 n t_s$ is also an angle expressed in radians. Maybe in an upcoming post we’ll be revisiting how the selection of $latex ts$ determines the quality of our sampling.

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