That is, if a sinusoid is input to an LTI system, the output will be a sinusoid at the same frequency, but possibly altered in amplitude and phase. This follows because the output of every LTI system can be expressed as a linear combination of delayed copies of the input signal. In this section, we derive this important result for the general case of sinusoids at the same frequency.
for and , given for . For each component sinusoid, we can write
Applying this expansion to Eq.(A.2) yields
where and are known. We now have two equations in two unknowns which are readily solved by (1) squaring and adding both sides to eliminate , and (2) forming a ratio of both sides of Eq.(A.4) to eliminate . The results are
phasor analysis that Eq.(A.2) always has a solution, we can express each component sinusoid as
reEquation (A.2) therefore becomes
Since is just the polar representation of a complex number, there is always some value of and such that equals whatever complex number results on the right-hand side of Eq.(A.5). As is often the case, we see that the use of Euler's identity and complex analysis gives a simplified algebraic proof which replaces a proof based on trigonometric identities.
Phasor Analysis: Factoring a Complex Sinusoid into Phasor Times CarrierThe heart of the preceding proof was the algebraic manipulation
reThus, a sinusoid is determined by its frequency (which specifies the carrier term) and its phasor , a complex constant. Phasor analysis is discussed further in .
Elementary Filter Sections
Complex and Trigonometric Identities