Another interesting notion arises from the commutative property of multiplication (exploited in an example above (Example 4.6)): We can rather arbitrarily choose an order in which to apply each product. Consider a cascade of two linear, time-invariant systems. Because the Fourier transform of the first system's output is X (f) H1 (f) and it serves as the second system's input, the cascade's output spectrum is X (f) H1 (f) H2 (f). Because this product also equals X (f) H2 (f) H1 (f), the cascade having the linear systems in the opposite order yields the same result. Furthermore, the cascade acts like a single linear system, having transfer function H1 (f) H2 (f). This result applies to other configurations of linear, time-invariant systems as well; see this Frequency Domain Problem (Problem 4.13). Engineers exploit this property by determining what transfer function they want, then breaking it down into components arranged according to standard confgurations. Using the fact that op-amp circuits can be connected in cascade with the transfer function equaling the product of its component's transfer function (see this analog signal processing problem (Problem 3.44)), we find a ready way of realizing designs. We now understand why op-amp implementations of transfer functions are so important.
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Commutative Transfer Functions