Types of Signals

A signal is a quantity that shows meaningful information about some process. A signal can be an electrical signal (voltage, current etc), physical (pressure, temperature etc). All these signals convey information about some process may it be the monitoring of the output of an electronic circuit or the temperature of a heater. All signals in nature are continuous in time and amplitude. Since computers cannot process continuous signals, the continuous signals are first converted in to discrete form i-e they exist at specific instants of time and amplitude. There are four categories of signals with respect to their possible nature.

1. Continuous Time-Continuous Amplitude (CT-CA) Signals

These signals can exist at all amplitudes and time instants. These signals exist in nature.

2. Continuous Time-Discrete Amplitude (CT-DA) Signals

These signals can only take up values predefined allowable amplitude values but exist at any time instant. In these signals, the continuous amplitude is mapped in to a finite number of predefined levels. This is potentially a many to one mapping in which multiple input values are set to the same output values. This mapping process is called Quantisation and these type of signals are also called Quantised signals. The number of allowed levels depends on the number of bits used for the quantiser (4 bits have been used in the figure below).

3. Discrete Time-Continuous Amplitude (DT-CA) Signals

As opposed to CT-DA signals these signals can take up any amplitude value but can only exist at specific instants of time. The time instants at which the signal can exist are usually uniformly distributed (same separation between two amplitude points). Each of the amplitude points is called a ‘sample’ which literally means taking a sample from the signal at the time instant. Hence this signal is also called a Sampled signal.

4. Discrete Time- Discrete Amplitude (DT-DA) Signals

The processes of sampling and quantisation are combined to form the Discrete Time-Discrete Amplitude signals. These signals can take only predefined amplitude values and exist at specific instants of time. These are the signals which are processed by computers and are also known as Digital signals.

The conversion from a CT-CA signal to a DT-DA signal is a three step process, sampling, quantisation and encoding.

i. Sampling

Sampling is a fundamental process of DSP. It is the first stage in the conversion of a continuous signal to a discrete one. Sampling consists of taking sample points from the continuous signal at regular intervals. The duration of the interval is called the Sampling Period and its inverse called the Sampling Frequency. The result of sampling is a DT-CA amplitude signals.

Mathematically, the sampling process can be thought of a signal being multiplied by an infinite sequence of unit impulses each delayed by successive multiples of the Sampling Period. Since an impulse is nonzero only at one instant of time and is zero elsewhere, it only ‘picks up’ the sample of the input signal at that time instant. This property is called Sifting. This is illustrated in the figure below which shows a sampled version of one period of a sine wave having a frequency of 1Hz.

In the figure shown above, a total of 51 samples have been taken from the original signal resulting in 51 samples. One question which arises here is the number of samples to take from the continuous signal. This is dictated by the Nyquist Theorem which states that a band limited continuous signal can be reconstructed from its samples if they are acquired at a rate equal to twice the maximum frequency (Nyquist rate) present in the continuous signal. In the time domain, this equals to taking two samples per period of the signal (compare that to the 51 samples we took for a period above!). Undersampling (sampling at a rate lower than the frequency dictated by Nyquist theorem) and Oversampling (sampling at a rate higher than the frequency dictated by Nyquist theorem) give rise to undesirable problems and thus are generally to be avoided. One last thing to point out here is that for sinusoidal signals like the sine wave above, we need to sample at a rate slightly higher than the Nyquist rate since the sampling points may fall on the zero crossings of the sinusoid.

ii. Quantisation

As mentioned before, quantisation is the mapping of infinitely possible continuous values in to a finite set of possible levels. The motivation for quantisation stems from the fact that computers work with a finite number of bits to represent data thus allowing them a finite range and precision. This requires that quantities in the real world are mapped in to a finite set of values with finite precision to be able to use computers etc to manipulate them. Generally, the range of the continuous signal is divided into segments called quantisation intervals, each segment is associated with a value called quantisation level. At any time, the quantised form of a continuous signal will be assigned the value of the quantisation level according to interval in which it falls. There are two main types of quantisation such as Scalar/Vector which can be of Uniform and Non-Uniform nature. We will consider the case of Uniform scalar quantisation as this type is generally used in Analog to Digital Converters (ADC) used in embedded systems. Uniform scalar quantisation involves quantisation intervals which are of the same size and quantisation levels which exist at the middle of each interval. This is illustrated in the figure below.

There are eight quantisation levels (a0 to a7) and nine quantisation intervals. For each quantisation interval, the quantisation level falls at its centre, except for the first and the last intervals. This means that any value of the continuous signal that falls below the considered range will be assigned the first quantisation level and so forth for the maximum case. One thing to note is that for a uniform quantiser as shown above, the quantisation error which is the difference between the quantised signal value and the original signal value can be no more than half the quantisation interval. The quantisation interval is also called the Resolution of the quantiser.

iii. Encoding

Once the signal has been converted in to a finite set of values, it needs to be represented in some appropriate manner. This process of representing the quantised signal is called Encoding. There are many ways to represent the quantised signal, however, for our consideration we take the case of simple binary coding in which each successive quantisation level is represented by a sequential binary number. For the figure shown above, the quantisation levels a0 to a7 can be encoded using three bits (B2,B1,B0) as,

This is the most common method of encoding quantisation levels, however, other coding techniques such as grey codes can also be used.