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.