## An Easier Way

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 [84]; 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].

### Complex Sinusoids

Using Euler's identity to represent sinusoids, we have

when time is continuous (see §A.1 for a list of notational conventions), and when time is discrete,

Any function of the form
or
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
either
*sine* or *cosine* separately. One may even take the
point of view that
is *simpler* and *more
fundamental* than
or
, as evidenced by
the following identities (which are immediate consequences of Euler's
identity,
Eq.(1.8)):

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 [84] and elsewhere [53,14]. Here, we will look only at some intuitive connections between complex sinusoids and the more familiar real sinusoids.

### Complex Amplitude

Note that the amplitude and phase can be viewed as the magnitude and angle of a single complex number

*complex amplitude*of the complex sinusoid defined by the left-hand side of either Eq.(1.9) or Eq.(1.10). The complex amplitude is the same whether we are talking about the continuous-time sinusoid or the discrete-time sinusoid .

### Phasor Notation

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.

### Plotting Complex Sinusoids as Circular Motion

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

Thus,

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 [84].

### 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

*cosine*goes in is the

*real*part of the signal, and the other signal path is simply called the

*imaginary*part. Thus, a complex signal in real life is implemented as two real signals processed in parallel; in particular, a complex sinusoid is implemented as two real sinusoids, side by side, one-quarter cycle out of phase. When the filter itself is real, two copies of it suffice to process a complex signal. If the filter is complex, we must implement complex multiplies between the complex signal samples and filter coefficients.

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

and

*cosine*varies, from 2 to 0 as the frequency of an input sinusoid goes from 0 to half the sampling rate. In other words, the amplitude response of Eq.(1.1) goes sinusoidally from 2 to 0 as goes from 0 to . It does seem somewhat reasonable to consider it a low-pass, and it is a poor one in the sense that it is hard to see which frequency should be called the cut-off frequency. We see that the spectral ``roll-off'' is very slow, as low-pass filters go, and this is what we pay for the extreme simplicity of Eq.(1.1). The phase response is linear in frequency, which gives rise to a constant time delay irrespective of the signal frequency.

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.

Tying it all together, the general output of a linear time-invariant filter with a complex sinusoidal input may be expressed as

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Summary

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Finding the Frequency Response