Commonly used operations on signals include transformations of amplitude or time:
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Amplitude scaling of \(x(t)\) by \(C\) multiplies all signal values by constant \(C\).
\(\boxed{x(t) \to C\,x(t)}\)
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An amplitude shift adds a constant \(K\) to \(x(t)\).
\(\boxed{x(t) \to K + x(t)}\)
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A time shift displaces a signal \(x(t)\) in time without changing its shape.
\(\boxed{x(t) \to x(t - \alpha)}\quad\textrm{(delayed, shifted right by }\alpha\textrm{)}\)
\(\boxed{x(t) \to x(t + \alpha)}\quad\textrm{(advanced, shifted left by }\alpha\textrm{)}\)
where \(\alpha > 0\)
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Time scaling speeds up or slows down time and results in signal compression or expansion.
\(\boxed{x(t) \to x(\alpha t)}\quad\textrm{(compressed, speeded up to }\alpha t\textrm{)}\)
\(\boxed{x(t) \to x\left(t / \alpha\right)}\quad\textrm{(expanded, slowed down to }t / \alpha\textrm{)}\)
where \(\alpha > 1\)
Note that shifting or folding a signal \(x(t)\) will not change its area or energy, but time scaling \(x(t)\) to \(x(\alpha t)\) will reduce both its area and energy by \(|\alpha|\).
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Reflection or folding or time reversal is just a scaling operation with \(\alpha = -1\). It creates the folded signal \(x(-t)\) as a mirror image to \(x(t)\) about the vertical axis passing through the origin \(t = 0\).
\(\boxed{x(t) \to x(-t)}\)
The signal \(y(t) = x(\alpha t - \beta)\) may be generated from \(x(t)\) by plotting \(x(t)\) against a new time axis \(t_n\).
\(\displaystyle x(t) = x(\alpha t_n - \beta) \to \boxed{t = \alpha t_n - \beta}\)
\(\boxed{t_n = \frac{1}{\alpha}(t + \beta)}\)
Combined Operations
The most general operation involving all the three operations (shift, scale, and reflect) is \(x(\alpha t - \beta)\), which is realized in two possible sequences of operation:
- Time shift \(x(t)\) by \(\beta\) to obtain \(x(t - \beta)\). Then time scale the shifted signal \(x(t - \beta)\) by \(\alpha\) to obtain \(x(\alpha t - \beta)\).
- Time scale \(x(t)\) by \(\alpha\) to obtain \(x(\alpha t)\). Then time shift the scaled signal \(x(\alpha t)\) by \(\beta / \alpha\) to obtain \(x[\alpha(t - \beta / \alpha)] = x(\alpha t - \beta)\).
In either case, if \(\alpha\) is negative, time scaling involves time reversal.