Basicoperations
As discussed previously, discrete signals are formed by sampling, quantising and encoding analogue signals. It is now important to talk about fundamental operations which can be performed on these signals. The three basic operations are scaling, shifting and reversal or reflection. All these operations can be performed on amplitude as well as in time.
1. Scaling
Scaling involves multiplying the amplitude or the time index of a signal by a number to change them in a suitable manner. Let a discrete signal be denoted by x[n].
An amplitude scaled version of the signal x[n] with a scaling factor of A can be written as
This results in the amplitude of the original multiplied by the constant factor A i-e equivalent to amplification by the factor A. A factor of greater than zero will increase the amplitude and vice versa. This is illustrated below for a sine wave of unit amplitude which has been scaled with a factor of 2 and ½.
Similarly the time scaled version of the signal x[n] with the scaled in time by a factor of a can be written as,
A value of a larger than one will result in the signal shrinking along the time axis and a value less than one will result in the signal expanding along the time axis. This is shown in the figure below
2. Shifting
The Shifting operation allows for signals to be shifted along the amplitude or the time axis. Shifting signals over the amplitude axis is synonymous to adding a DC offset to the signal. As can be expected, adding a negative offset will reduce the highest positive value in the signal and vice versa. An amplitude shifted version of a signal x[n] by an amount Δx can be written as
The figure below shows a sine wave shifted by a positive value of 2 and a negative value of 2.
Shifting along the time axis can either delay or advance the samples. A shifted version of a signal x[n] by an amount Δn (Δn is the number of samples) can be written as
If Δn is positive, the signal samples are moved to the right and the process is called a Time Delay, when Δn is negative, the signal samples are moved to the left and the process is called Time advance. This has been shown for a sine wave below
The operation of time shifting is widely used in digital filters in which a ‘tap delay line’ is created.
3. Reversal or Reflection
Reversal is the process of mirroring a signal. Like the Scaling and Shifting operations, reversal can be performed along the amplitude as well as the time axis. An amplitude reversed signal x[n] can be written as
This results in a signal which is mirrored along the amplitude axis and is shown for a sine wave in the figure below
Similarly, a signal can be reversed along the time axis. For a signal x[n], this can be written as
A time reversed version of a sine wave is shown below
The operation of time reversal is used in the process of convolution as will be discussed in upcoming sections.