We derived the frequency response above using trig identities in order to minimize the mathematical level involved. However, it turns out it is actually easier, though more advanced, to use complex numbers for this purpose. To do this, we need Euler's identity:
where is the imaginary unit for complex numbers, and is a transcendental constant approximately equal to . Euler's identity is fully derived in ; here we will simply use it ``on faith.'' It can be proved by computing the Taylor series expansion of each side of Eq.(1.8) and showing equality term by term [84,14].
when time is continuous (see §A.1 for a list of notational conventions), and when time is discrete,
Any function of the form
will henceforth be called a complex
sinusoid.2.3 We will
see that it is easier to manipulate both sine and
cosine simultaneously in this form than it is to deal with
sine or cosine separately. One may even take the
point of view that
is simpler and more
, as evidenced by
the following identities (which are immediate consequences of Euler's
Thus, sine and cosine may each be regarded as a combination of two complex sinusoids. Another reason for the success of the complex sinusoid is that we will be concerned only with real linear operations on signals. This means that in Eq.(1.8) will never be multiplied by or raised to a power by a linear filter with real coefficients. Therefore, the real and imaginary parts of that equation are actually treated independently. Thus, we can feed a complex sinusoid into a filter, and the real part of the output will be the cosine response and the imaginary part of the output will be the sine response. For the student new to analysis using complex variables, natural questions at this point include ``Why ?, Where did the imaginary exponent come from? Are imaginary exponents legal?'' and so on. These questions are fully answered in  and elsewhere [53,14]. Here, we will look only at some intuitive connections between complex sinusoids and the more familiar real sinusoids.
Note that the amplitude and phase can be viewed as the magnitude and angle of a single complex number
The complex amplitude is also defined as the phasor associated with any sinusoid having amplitude and phase . The term ``phasor'' is more general than ``complex amplitude'', however, because it also applies to the corresponding real sinusoid given by the real part of Equations (1.9-1.10). In other words, the real sinusoids and may be expressed as
and is the associated phasor in each case. Thus, we say that the phasor representation of is . Phasor analysis is often used to analyze linear time-invariant systems such as analog electrical circuits.
Figure 1.8 shows Euler's relation graphically as it applies to sinusoids. A point traveling with uniform velocity around a circle with radius 1 may be represented by in the complex plane, where is time and is the number of revolutions per second. The projection of this motion onto the horizontal (real) axis is , and the projection onto the vertical (imaginary) axis is . For discrete-time circular motion, replace by to get which may be interpreted as a point which jumps an arc length radians along the circle each sampling instant.
For circular motion to ensue, the sinusoidal motions must be at the same frequency, one-quarter cycle out of phase, and perpendicular (orthogonal) to each other. (With phase differences other than one-quarter cycle, the motion is generally elliptical.)
The converse of this is also illuminating. Take the usual circular motion which spins counterclockwise along the unit circle as increases, and add to it a similar but clockwise circular motion . This is shown in Fig.1.9. Next apply Euler's identity to get
This statement is a graphical or geometric interpretation of Eq.(1.11). A similar derivation (subtracting instead of adding) gives the sine identity Eq.(1.12).
We call a positive-frequency sinusoidal component when , and is the corresponding negative-frequency component. Note that both sine and cosine signals have equal-amplitude positive- and negative-frequency components (see also [84,53]). This happens to be true of every real signal (i.e., non-complex). To see this, recall that every signal can be represented as a sum of complex sinusoids at various frequencies (its Fourier expansion). For the signal to be real, every positive-frequency complex sinusoid must be summed with a negative-frequency sinusoid of equal amplitude. In other words, any counterclockwise circular motion must be matched by an equal and opposite clockwise circular motion in order that the imaginary parts always cancel to yield a real signal (see Fig.1.9). Thus, a real signal always has a magnitude spectrum which is symmetric about 0 Hz. Fourier symmetries such as this are developed more completely in .
Rederiving the Frequency Response
Let's repeat the mathematical sine-wave analysis of the simplest low-pass filter, but this time using a complex sinusoid instead of a real one. Thus, we will test the filter's response at frequency by setting its input to
Using the normal rules for manipulating exponents, we find that the output of the simple low-pass filter in response to the complex sinusoid at frequency Hz is given by
where we have defined , which we will show is in fact the frequency response of this filter at frequency . This derivation is clearly easier than the trigonometry approach. What may be puzzling at first, however, is that the filter is expressed as a frequency-dependent complex multiply (when the input signal is a complex sinusoid). What does this mean? Well, the theory we are blindly trusting at this point says it must somehow mean a gain scaling and a phase shift. This is true and easy to see once the complex filter gain is expressed in polar form,
It is now easy to see that
It deserves to be emphasized that all a linear time-invariant filter can do to a sinusoid is scale its amplitude and change its phase. Since a sinusoid is completely determined by its amplitude , frequency , and phase , the constraint on the filter is that the output must also be a sinusoid, and furthermore it must be at the same frequency as the input sinusoid. More explicitly:
Mathematically, a sinusoid has no beginning and no end, so there really are no start-up transients in the theoretical setting. However, in practice, we must approximate eternal sinusoids with finite-time sinusoids whose starting time was so long ago that the filter output is essentially the same as if the input had been applied forever.
Finding the Frequency Response