A Sum of Sinusoids at the Same Frequency is Another Sinusoid at that Frequency

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A Sum of Sinusoids at the
Same Frequency is Another
Sinusoid at that Frequency

It is an important and fundamental fact that a sum of sinusoids at the same frequency, but different phase and amplitude, can always be expressed as a single sinusoid at that frequency with some resultant phase and amplitude. An important implication, for example, is that

$\textstyle \parbox{0.8\textwidth}{sinusoids are eigenfunctions of linear time-invariant (LTI) systems.}$

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.

Proof Using Trigonometry

We want to show it is always possible to solve

$\displaystyle A\cos(\omega t + \phi) = A_1\cos(\omega t + \phi_1) + A_2\cos(\omega t + \phi_2) + \cdots + A_N\cos(\omega t + \phi_N) \protect$

(A.2)

for

and $\phi$ , given $A_i, \phi_i$ for $i=1,\ldots,N$ . For each component sinusoid, we can write

$\displaystyle A_i\cos(\omega t + \phi_i)$	$\displaystyle =$	$\displaystyle A_i\cos(\omega t)\cos(\phi_i) - A_i\sin(\omega t)\sin(\phi_i)$
	$\displaystyle =$	$\displaystyle \left[A_i\cos(\phi_i)\right]\cos(\omega t) - \left[A_i\sin(\phi_i)\right]\sin(\omega t)$	(A.3)

Applying this expansion to Eq. $\,$ (A.2) yields

$\begin{eqnarray*} \left[A\cos(\phi)\right]\cos(\omega t) &-&\left[A\sin(\phi)\ri... ...a t) - \left[\sum_{i=1}^N A_i\sin(\phi_i)\right]\sin(\omega t). \end{eqnarray*}$

Equating coefficients gives

$\displaystyle A\cos(\phi)$	$\displaystyle =$	$\displaystyle \sum_{i=1}^N A_i\cos(\phi_i) \isdefs x$
$\displaystyle A\sin(\phi)$	$\displaystyle =$	$\displaystyle \sum_{i=1}^N A_i\sin(\phi_i) \isdefs y. \protect$	(A.4)

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 $\phi$ , and (2) forming a ratio of both sides of Eq. $\,$ (A.4) to eliminate

. The results are

$\begin{eqnarray*} A &=& \sqrt{x^2+y^2}\\ \phi &=& \tan^{-1}\left(\frac{y}{x}\right) \end{eqnarray*}$

which has a unique solution for any values of and $\phi_i$ .

Proof Using Complex Variables

To show by means of phasor analysis that Eq. $\,$ (A.2) always has a solution, we can express each component sinusoid as

$\displaystyle A_i\cos(\omega t + \phi_i) =$ re $\displaystyle \left\{A_i e^{j(\omega t + \phi_i)}\right\}$

Equation (A.2) therefore becomes

$\begin{eqnarray*} \mbox{re}\left\{A e^{j(\omega t + \phi)}\right\} &=& \sum_{i=1... ...}\right\}\\ &=& \mbox{re}\left\{A e^{j(\omega t+\phi)}\right\}. \end{eqnarray*}$

Thus, equality holds when we define

$\displaystyle A e^{j\phi} \isdef \sum_{i=1}^N A_i e^{j\phi_i}. \protect$

(A.5)

Since $A e^{j\phi}$ is just the polar representation of a complex number, there is always some value of $A\geq 0$ and $\phi\in[-\pi,\pi)$ such that $A e^{j\phi}$ 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 Carrier

The heart of the preceding proof was the algebraic manipulation

$\displaystyle \sum_{i=1}^N A_i e^{j(\omega t + \phi_i)} = e^{j\omega t} \sum_{i=1}^N A_i e^{j\phi_i}.$

The carrier term $e^{j\omega t}$ ``factors out'' of the sum. Inside the sum, each sinusoid is represented by a complex constant $A_i e^{j\phi_i}$ , known as the phasor associated with that sinusoid.

For an arbitrary sinusoid having amplitude , phase $\phi$ , and radian frequency $\omega$ , we have

$\displaystyle A\cos(\omega t + \phi) =$ re $\displaystyle \left\{(A e^{j\phi}) e^{j\omega t}\right\}.$

Thus, a sinusoid is determined by its frequency $\omega$ (which specifies the carrier term) and its phasor ${\cal A}\isdef A e^{j\phi}$ , a complex constant. Phasor analysis is discussed further in [84].

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