13.9:
Basic Operations on Signals
Basic signal operations are time reversal, scaling, shifting, and amplitude transformations.
Time reversal mirrors a continuous-time signal about the vertical axis at time equals zero, achieved by substituting 't' with negative 't'. For a considered signal, the results are shown graphically.
Time scaling compresses or expands a signal in time by replacing 't' with 'at', where 'a' is constant. If the magnitude of this constant is greater than 1, the signal compresses; if it's less than 1, it expands.
A negative value of this constant induces both time reversal and compression or expansion. This can be graphically represented using the considered signal.
Time shifting of a continuous-time signal is done by replacing 't' with 't − t0', where 't0' is constant. A positive constant delays and shifts the signal right from the origin, while a negative one advances the signal and shifts it left.
Amplitude transformations of a continuous-time signal take the general form where 'A' and 'B' are constants. The graph shows an amplitude transformation of an exemplified signal.
13.9:
Basic Operations on Signals
Basic signal operations include time reversal, time scaling, time shifting, and amplitude transformations. These operations are fundamental in signal processing and analysis.
Time Reversal mirrors a continuous-time signal about the vertical axis at t=0. This is achieved by substituting t with −t. For example, if a signal x(t) is considered, the time-reversed signal is x(−t). This operation can be graphically represented, showing the mirrored signal.
Time Scaling compresses or expands a signal in time by replacing t with at, where a is a constant. If ∣a∣>1, the signal compresses; if ∣a∣<1, it expands. A negative value of a induces both time reversal and compression or expansion. A graphical representation of this operation on a signal demonstrates these effects.
Time Shifting of a continuous-time signal is done by replacing t with t−t0, where t0 is a constant. A positive t0 delays and shifts the signal to the right, while a negative t0 advances the signal and shifts it to the left. For instance, a signal x(t) shifted by t0 is represented as x(t−t0).
Amplitude Transformations of a continuous-time signal are generally of the form Ax(t)+B, where A and B are constants. These transformations scale the amplitude and can also shift it vertically. Graphically, an amplitude transformation of a signal shows the effect of scaling by A and shifting by B.
These basic operations—time reversal, scaling, shifting, and amplitude transformations—are essential tools for manipulating and analyzing signals, allowing for various adjustments and modifications in signal processing applications.