Basic Signal Operations
A signal, comprises of a set of information expressed as a function of any number of independent variables, that can be given as an input to a system, or derived as output from the system, to realize its true practical utility. The signal we derive out of a complex system might not always be in the form we want,
∴ being well acquainted with some basic signal operations may come really handy to enhance the understandability and applicability of signals. The mathematical transformation from one signal to another can be expressed as Where, Y(t) represents the modified signal derived from the original signal X(t), having only one independent variable t. The basic set of signal operations can be broadly classified as below.
Basic Signal Operations Performed on Dependent VariablesIn this transformation, only the quadrature axis values are modified i.e magnitude of the signal changes, with no effects on the horizontal axis values or periodicity of signals like.
- Amplitude scaling of signals.
- Addition of signals.
- Multiplication of signals.
- Differentiation of signals.
- Integration of signals.
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Amplitude Scaling of SignalsAmplitude scaling is a very basic operation performed on signals to vary its strength. It can be mathematically represented as Y(t) = α X(t). Here, α is the scaling factor, where:- α<1 → signal is attenuated. α>1 → signal is amplified. This is illustrated in the diagram, where the signal is attenuated when α = 0.5 in fig (b) and amplified when α = 1.5 as in fig (c).
Addition of SignalsThis particular operation involves the addition of amplitude of two or more signals at each instance of time or any other independent variables which are common between the signals. Addition of signals is illustrated in the diagram below, where X1(t) and X2(t) are two time dependent signals, performing the additional operation on them we get,
Multiplication of SignalsLike addition multiplication of signals also falls under the category of basic signal operations. Here multiplication of amplitude of two or more signals at each instance of time or any other independent variables is done which are common between the signals. The resultant signal we get has values equal to the product of amplitude of the parent signals for each instance of time. Multiplication of signals is illustrated in the diagram below, where X1(t) and X2(t) are two time dependent signals, on whom after performing the multiplication operation we get,
Differentiation of SignalsFor differentiation of signals, it must be noted that this operation is only applicable for only continuous signals, as a discrete function cannot be differentiated. The modified signal we get on differentiation has tangential values of the parent signal at all instance of time. Mathematically it can be expressed as:- Differentiation of a standard square and sine wave is shown in the figure below.
Integration of SignalsLike differentiation, integration of signals is also applicable to only continuous time signals. The limits of integration will be from - ∞ to present instance of time t. It is mathematically expressed as, Integration of some continuous time signals is shown in the diagram below.
Basic Signal Operations Performed on Dependent VariablesThis is exactly the opposite of the above mentioned case, here the periodicity of the signal is varied by modifying the horizontal axis values, while the amplitude or the strength remains constant. These are:-
- Time scaling of signals
- Reflection of signals
- Time-shifting of signals.