Discrete systems and signal properties
This sections deals with useful properties of digital signals and introduces the Digital Signal Processing (DSP) System.
1. Signal Properties
A signal can be characterised by several properties which can be used to identify signal traits. Some of the fundamental properties are discussed below.
i. Even and Odd Signals
A signal x[n] is said to be an Even signal if it is symmetric along the time/sample axis i-e
If the above condition does not hold true, the signal is called an Odd signal. An example of an Even signal is a cosine wave and an Odd signal example is a sine wave as shown below
ii. Periodic and Aperiodic Signals
Periodic signals repeat their amplitude values at regular intervals. Mathematically, a signal is said to be a periodic signal if it satisfies the following equation
where N is the period of the signal. A signal for which there is no value of N that satisfies the above equation is called an Aperiodic signal. Examples of both signal types are shown in the figure below
iii. Causal and Non-Causal Signals
Causal signals are characterised by having zero value for all values of negative time, mathematically
On the other hand, Anti-Causal signals are characterised by having zero value for all positive time i-e
This is shown in the figure below
iv. Deterministic and Random Signals
A signal is said to be Deterministic if it can be written in a mathematical form and thus can be estimated. Examples of deterministic signals range from simple sinusoids to complex non-linear functions. Random signals are signals which cannot be described by a mathematical form and hence cannot be predicted. The figure below shows an example of both types
v. Sifting
The Sifting property is one of the most important properties in Digital Signal Processing. It formulates the process of sampling in much the same way as was discussed in the section describing the types of signals. The sifting property states that the multiplication of a signal and an impulse results in the value of the signal at the location of that impulse weighted by the magnitude of that impulse. Mathematically, it can be written as
Where x[n] is the signal under consideration, is an impulse of amplitude a at the index k. The sifting property is fundamental to evaluating responses of DSP systems to signals as it serves as the basis for the understanding of the convolution process.
2. DSP Systems
Discrete signals are generally processed by a DSP system to extract some useful information from the data or to change signals in some manner. These functions can range from a trivial application of filtering noise from sound to voice recognition and more. A DSP system is usually represented by a linear equation of the form
or more generally
Here y[n-k] are the system outputs at time index k (from 0→N), similarly, x[n-k] are the input samples at time index k (from 0→M). For a realisable system, M should always be less than N. It should be noted from the above equation that the output of a DSP system at any time is a weighted sum of the input and or output samples. Also, the equation of a DSP system is more ubiquitously represented using the z (delay) operator. Since we haven’t introduced the Z transform, this will be brought up later.
Before moving on to the next section, it is important to mention causality and non-causality in DSP systems. A Causal system is one for which the output does not depend on the future values of the input. On the other hand, a Non-Causal system has an output which depends on the future values of the input (time advanced versions of the input samples). Only Causal systems can be implemented as a real system on DSP hardware.