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Exponential and Sinusoidal Signals

This is a continuation from the previous tutorial - transformations of the independent variable.


In this tutorial and the next, we introduce several basic continuous-time and discrete-time signals. Not only do these signals occur frequently, but they also serve as basic building blocks from which we can construct many other signals.


1. Continuous-Time Complex Exponential and Sinusoidal Signals

The continuous-time complex exponential signal is of the form


where \(C\) and \(a\) are, in general, complex numbers.

Depending upon the values of these parameters, the complex exponential can exhibit several different characteristics.


Real Exponential Signals

As illustrated in Figure 1.19, if \(C\) and \(a\) are real [in which case \(x(t)\) is called a real exponential], there are basically two types of behavior.

If \(a\) is positive, then as \(t\) increases \(x(t)\) is a growing exponential, a form that is used in describing many different physical processes, including chain reactions in atomic explosions and complex chemical reactions.

If \(a\) is negative, then \(x(t)\) is a decaying exponential, a signal that is also used to describe a wide variety of phenomena, including the process of radioactive decay and the responses of RC circuits and damped mechanical systems.

Also, we note that for \(a=0\), \(x(t)\) is constant.


Figure 1.19   Continuous-time real exponential \(x(t)=Ce^{at}\): (a) \(a\gt0\); (b) \(a\lt0\).



Periodic Complex Exponential and Sinusoidal Signals

A second important class of complex exponentials is obtained by constraining \(a\) to be purely imaginary. Specifically, consider


An important property of this signal is that it is periodic.

To verify this, we recall from eq. (1.11) that \(x(t)\) will be periodic with period \(T\) if


Or, since


if follows that for periodicity, we must have


If \(\omega_0=0\), then \(x(t)=1\), which is periodic for any value of \(T\). If \(\omega_0\ne0\), then the fundamental period \(T_0\) of \(x(t)\) -- that is, the smallest positive value of \(T\) for which eq. (1.23) holds -- is


Thus, the signals \(e^{j\omega_0t}\) and \(e^{-j\omega_0t}\) have the same fundamental period.

A signal closely related to the periodic complex exponential is the sinusoidal signal


as illustrated in Figure 1.20.


Figure 1.20   Continuous-time sinusoidal signal.


With seconds as the units of \(t\), the units of \(\phi\) and \(\omega_0\) are radians and radians per second, respectively. It is also common to write \(\omega_0=2\pi{f_0}\), where \(f_0\) has the units of cycles per second, or hertz (Hz).

Like the complex exponential signal, the sinusoidal signal is periodic with fundamental period \(T_0\) given by eq. (1.24).

Sinusoidal and complex exponential signals are also used to describe the characteristics of many physical processes-in particular, physical systems in which energy is conserved.

For example, the natural response of an LC circuit is sinusoidal, as is the simple harmonic motion of a mechanical system consisting of a mass connected by a spring to a stationary support. The acoustic pressure variations corresponding to a single musical tone are also sinusoidal.

By using Euler's relation, the complex exponential in eq. (1.21) can be written in
terms of sinusoidal signals with the same fundamental period:


Similarly, the sinusoidal signal of eq. (1.25) can be written in terms of periodic complex exponentials, again with the same fundamental period:


Note that the two exponentials in eq. (1.27) have complex amplitudes.

Alternatively, we can express a sinusoid in terms of a complex exponential signal as


where, if \(c\) is a complex number, \(\mathcal{Re}\left\{c\right\}\) denotes its real part.

We will also use the notation \(\mathcal{Im}\left\{c\right\}\) for the imaginary part of \(c\), so that, for example,


From eq. (1.24), we see that the fundamental period \(T_0\) of a continuous-time sinusoidal signal or a periodic complex exponential is inversely proportional to \(|\omega_0|\), which we will refer to as the fundamental frequency.

From Figure 1.21, we see graphically what this means. If we decrease the magnitude of \(\omega_0\), we slow down the rate of oscillation and therefore increase the period. Exactly the opposite effects occur if we increase the magnitude of \(\omega_0\).

Consider now the case \(\omega_0=0\). In this case, as we mentioned earlier, \(x(t)\) is constant and therefore is periodic with period \(T\) for any positive value of \(T\). Thus, the fundamental period of a constant signal is undefined.

On the other hand, there is no ambiguity in defining the fundamental frequency of a constant signal to be zero. That is, a constant signal has a zero rate of oscillation.


Figure 1.21   Relationship between the fundamental frequency and period for continuous-time sinusoidal signals; here, \(\omega_1\gt\omega_2\gt\omega_3\), which implies that \(T_1\lt{T_2}\lt{T_3}\).


Periodic signals-and in particular, the complex periodic exponential signal in
eq. (1.21) and the sinusoidal signal in eq. (1.25) -- provide important examples of signals with infinite total energy but finite average power.

For example, consider the periodic exponential signal of eq. (1.21), and suppose that we calculate the total energy and average power in this signal over one period:



Since there are an infinite number of periods as \(t\) ranges from \(-\infty\) to \(\infty\), the total energy integrated over all time is infinite.

However, each period of the signal looks exactly the same. Since the average power of the signal equals \(1\) over each period, averaging over multiple periods always yields an average power of \(1\).

That is, the complex periodic exponential signal has finite average power equal to


Periodic complex exponentials will play a central role in much of our treatment of signals and systems, in part because they serve as extremely useful building blocks for many other signals.

We will often find it useful to consider sets of harmonically related complex exponentials-that is, sets of periodic exponentials, all of which are periodic with a common period \(T_0\).

Specifically, a necessary condition for a complex exponential \(e^{j\omega{t}}\) to be periodic with period \(T_0\) is that


which implies that \(\omega{T_0}\) is a multiple of \(2\pi\), i.e.,


Thus, if we define


we see that, to satisfy eq. (1.34), \(\omega\) must be an integer multiple of \(\omega_0\). That is, a harmonically related set of complex exponentials is a set of periodic exponentials with fundamental frequencies that are all multiples of a single positive frequency \(\omega_0\):


For \(k=0\), \(\phi_k(t)\) is a constant, while for any other value of \(k\), \(\phi_k(t)\) is periodic with fundamental frequency \(|k|\omega_0\) and fundamental period


The \(k\)th harmonic \(\phi_k(t)\) is still periodic with period \(T_0\) as well, as it goes through exactly \(|k|\) of its fundamental periods during any time interval of length \(T_0\).

Our use of the term "harmonic" is consistent with its use in music, where it refers to tones resulting from variations in acoustic pressure at frequencies that are integer multiples of a fundamental frequency.

For example, the pattern of vibrations of a string on an instrument such as a violin can be described as a superposition-i.e., a weighted sum--of harmonically related periodic exponentials.

In later tutorials, we will see that we can build a very rich class of periodic signals using the harmonically related signals of eq. (1.36) as the building blocks.


Example 1.5

It is sometimes desirable to express the sum of two complex exponentials as the product of a single complex exponential and a single sinusoid. For example, suppose we wish to plot the magnitude of the signal


To do this, we first factor out a complex exponential from the right side of eq. (1.38), where the frequency of this exponential factor is taken as the average of the frequencies of the two exponentials in the sum. Doing this, we obtain


which, because of Euler's relation, can be rewritten as


From this, we can directly obtain an expression for the magnitude of \(x(t)\):


Here, we have used the fact that the magnitude of the complex exponential \(e^{j2.5t}\) is always unity. Thus, \(|x(t)|\) is what is commonly referred to as a full-wave rectified sinusoid, as shown in Figure 1.22.


Figure 1.22  The full-wave rectified sinusoid of Example 1.5.



General Complex Exponential Signals

The most general case of a complex exponential can be expressed and interpreted in terms of the two cases we have examined so far: the real exponential and the periodic complex exponential.

Specifically, consider a complex exponential \(Ce^{at}\), where \(C\) is expressed in polar form and \(a\) in rectangular form. That is,






Using Euler's relation, we can expand this further as


Thus, for \(r=0\), the real and imaginary parts of a complex exponential are sinusoidal.

For \(r\gt0\) they correspond to sinusoidal signals multiplied by a growing exponential, and for \(r\lt0\) they correspond to sinusoidal signals multiplied by a decaying exponential. These two cases are shown in Figure 1.23.

The dashed lines in the figure correspond to the functions \(\pm|C|e^{rt}\). From eq. (1.42), we see that \(|C|e^{rt}\) is the magnitude of the complex exponential: Thus, the dashed curves act as an envelope for the oscillatory curve in the figure in that the peaks of the oscillations just reach these curves, and in this way the envelope provides us with a convenient way to visualize the general trend in the amplitude of the oscillations.


Figure 1.23.  (a) Growing sinusoidal signal \(x(t)=Ce^{rt}\cos(\omega_0t+\theta)\), \(r\gt0\);  (b) decaying sinusoid \(x(t)=Ce^{rt}\cos(\omega_0t+\theta)\), \(r\lt0\).


Sinusoidal signals multiplied by decaying exponentials are commonly referred to as damped sinusoids. Examples of damped sinusoids arise in the response of RLC circuits and in mechanical systems containing both damping and restoring forces, such as automotive suspension systems.

These kinds of systems have mechanisms that dissipate energy (resistors, damping forces such as friction) with oscillations that decay in time.



2. Discrete-Time Complex Exponential and Sinusoidal Signals

As in continuous time, an important signal in discrete time is the complex exponential signal or sequence, defined by


where \(C\) and \(\alpha\) are, in general, complex numbers.

This could alternatively be expressed in the form




Although the form of the discrete-time complex exponential sequence given in eq. (1.45) is more analogous to the form of the continuous-time exponential, it is often more convenient to express the discrete-time complex exponential sequence in the form of eq. (1.44).


Real Exponential Signals

If \(C\) and \(\alpha\) are real, we can have one of several types of behavior, as illustrated in Figure 1.24.

If \(|\alpha|\gt1\) the magnitude of the signal grows exponentially with \(n\), while if \(|\alpha|\lt1\) we have a decaying exponential. Furthermore, if \(\alpha\) is positive, all the values of \(C\alpha^n\) are of the same sign, but if \(\alpha\) is negative then the sign of \(x[n]\) alternates.

Note also that if \(\alpha=1\) then \(x[n]\) is a constant, whereas if \(\alpha=-1\), \(x[n]\) alternates in value between \(+C\) and \(-C\).

Real-valued discrete-time exponentials are often used to describe population growth as a function of generation and total return on investment as a function of day, month, or quarter.


Figure 1.24.  The real exponential signal \(x[n]=C\alpha^n\): (a) \(\alpha\gt1\);  (b) \(0\lt\alpha\lt1\);  (c) \(-1\lt\alpha\lt0\);  (d) \(\alpha\lt-1\).



Sinusoidal Signals

Another important complex exponential is obtained by using the form given in eq. (1.45) and by constraining \(\beta\) to be purely imaginary (so that \(|\alpha|=1\). Specifically, consider


As in the continuous-time case, this signal is closely related to the sinusoidal signal


If we take \(n\) to be dimensionless, then both \(\omega_0\) and \(\phi\) have units of radians. Three examples of sinusoidal sequences are shown in Figure 1.25.


Figure 1.25.  Discrete-time sinusoidal signals.


As before, Euler's relation allows us to relate complex exponentials and sinusoids:




The signals in eqs. (1.46) and (1.47) are examples of discrete-time signals with infinite total energy but finite average power. For example, since \(|e^{j\omega_0n}|^2=1\), every sample of the signal in eq. (1.46) contributes \(1\) to the signal's energy. Thus, the total energy for \(-\infty\lt{n}\lt\infty\) is infinite, while the average power per time point is obviously equal to \(1\).


General Complex Exponential Signals

The general discrete-time complex exponential can be written and interpreted in terms of real exponentials and sinusoidal signals. Specifically, if we write \(C\) and \(\alpha\) in polar form,







Thus, for \(|\alpha|=1\), the real and imaginary parts of a complex exponential sequence are sinusoidal.

For \(|\alpha|\lt1\) they correspond to sinusoidal sequences multiplied by a decaying exponential, while for \(|\alpha|\gt1\) they correspond to sinusoidal sequences multiplied by a growing exponential.

Examples of these signals are depicted in Figure 1.26.


Figure 1.26.  (a) Growing discrete-time sinusoidal signals;  (b) decaying discrete-time sinusoid.



3. Periodicity Properties of Discrete-Time Complex Exponentials

While there are many similarities between continuous-time and discrete-time signals, there are also a number of important differences.

One of these concerns the discrete-time exponential signal \(e^{j\omega_0n}\). In part 1 above, we identified the following two properties of its continuous-time counterpart \(e^{j\omega_0t}\): (1) the larger the magnitude of \(\omega_0\), the higher is the rate of oscillation in the signal; and (2) \(e^{j\omega_0t}\) is periodic for any value of \(\omega_0\).

In this section we describe the discrete-time versions of both of these properties, and as we will see, there are definite differences between each of these and its continuous-time counterpart.

The fact that the first of these properties is different in discrete time is a direct consequence of another extremely important distinction between discrete-time and continuous-time complex exponentials.

Specifically, consider the discrete-time complex exponential with frequency \(\omega_0+2\pi\):


From eq. (1.51), we see that the exponential at frequency \(\omega_0+2\pi\) is the same as that at frequency \(\omega_0\). Thus, we have a very different situation from the continuous-time case, in which the signals \(e^{j\omega_0t}\) are all distinct for distinct values of \(\omega_0\).

In discrete time, these signals are not distinct, as the signal with frequency \(\omega_0\) is identical to the signals with frequencies \(\omega_0\pm2\pi\), \(\omega_0\pm4\pi\), and so on.

Therefore, in considering discrete-time complex exponentials, we need only consider a frequency interval of length \(2\pi\) in which to choose \(\omega_0\). Although, according to eq. (1.51), any interval of length \(2\pi\) will do, on most occasions we will use the interval \(0\le\omega_0\lt2\pi\) or the interval \(-\pi\le\omega_0\lt\pi\).

Because of the periodicity implied by eq. (1.51), the signal \(e^{j\omega_0n}\) does not have a continually increasing rate of oscillation as \(\omega_0\) is increased in magnitude.

Rather, as illustrated in Figure 1.27, as we increase \(\omega_0\) from \(0\), we obtain signals that oscillate more and more rapidly until we reach \(\omega_0=\pi\). As we continue to increase \(\omega_0\), we decrease the rate of oscillation until we reach \(\omega_0=2\pi\), which produces the same constant sequence as \(\omega_0=0\).

Therefore, the low-frequency (that is, slowly varying) discrete-time exponentials have values of \(\omega_0\) near \(0\), \(2\pi\), and any other even multiple of \(\pi\), while the high frequencies (corresponding to rapid variations) are located near \(\omega_0=\pm\pi\) and other odd multiples of \(\pi\).

Note in particular that for \(\omega_0=\pi\) or any other odd multiple of \(\pi\),


so that this signal oscillates rapidly, changing sign at each point in time [as illustrated in Figure 1.27(e)].


Figure 1.27  Discrete-time sinusoidal sequences for several different frequencies.


The second property we wish to consider concerns the periodicity of the discrete-time complex exponential. In order for the signal \(e^{j\omega_0n}\) to be periodic with period \(N\gt0\), we must have


or equivalently,


For eq. (1.54) to hold, \(\omega_0N\) must be a multiple of \(2\pi\). That is, there must be an integer \(m\) such that


or equivalently,


According to eq. (1.56), the signal \(e^{j\omega_0n}\) is periodic if \(\omega_0/2\pi\) is a rational number and is not periodic otherwise.

These same observations also hold for discrete-time sinusoids. For example, the signals depicted in Figure 1.25(a) and (b) are periodic, while the signal in Figure 1.25(c) is not.

Using the calculations that we have just made, we can also determine the fundamental period and frequency of discrete-time complex exponentials, where we define the fundamental frequency of a discrete-time periodic signal as we did in continuous time. That is, if \(x[n]\) is periodic with fundamental period \(N\), its fundamental frequency is \(2\pi/N\).

Consider, then, a periodic complex exponential \(x[n]=e^{j\omega_0n}\) with \(\omega_0\ne0\). As we have just seen, \(\omega_0\) must satisfy eq. (1.56) for some pair of integers \(m\) and \(N\), with \(N\gt0\).

It is shown that if \(\omega_0\ne0\) and if \(N\) and \(m\) have no factors in common, then the fundamental period of \(x[n]\) is \(N\). Using this fact together with eq. (1.56), we find that the fundamental frequency of the periodic signal \(e^{j\omega_0n}\) is


Note that the fundamental period can also be written as


These last two expressions again differ from their continuous-time counterparts. In Table 1.1, we have summarized some of the differences between the continuous-time signal \(e^{j\omega_0t}\) and the discrete-time signal \(e^{j\omega_0n}\).

Note that, as in the continuous-time case, the constant discrete-time signal resulting from setting \(\omega_0=0\) has a fundamental frequency of zero, and its fundamental period is undefined.


Table 1.1   Comparison of the signals \(e^{j\omega_0t}\) and \(e^{j\omega_0n}\).


To gain some additional insight into these properties, let us examine again the signals depicted in Figure 1.25.

First, consider the sequence \(x[n]=\cos(2\pi{n}/12)\), depicted in Figure 1.25(a), which we can think of as the set of samples of the continuous-time sinusoid \(x(t)=\cos(2\pi{t}/12)\) at integer time points. In this case, \(x(t)\) is periodic with fundamental period 12 and \(x[n]\) is also periodic with fundamental period 12. That is, the values of \(x[n]\) repeat every 12 points, exactly in step with the fundamental period of \(x(t)\).

In contrast, consider the signal \(x[n]=\cos(8\pi{n}/31)\), depicted in Figure 1.25(b), which we can view as the set of samples of \(x(t)=\cos(8\pi{t}/31)\) at integer points in time. In this case, \(x(t)\) is periodic with fundamental period \(31/4\). On the other hand, \(x[n]\) is periodic with fundamental period \(31\).

The reason for this difference is that the discrete-time signal is defined only for integer values of the independent variable. Thus, there is no sample at time \(t=31/4\), when \(x(t)\) completes one period (starting from \(t=0\)). Similarly, there is no sample at \(t=2\cdot31/4\) or \(t=3\cdot31/4\), when \(x(t)\) has completed two or three periods, but there is a sample at \(t=4\cdot31/4=31\), when \(x(t)\) has completed four periods. This can be seen in Figure 1.25(b), where the pattern of \(x[n]\) values does not repeat with each single cycle of positive and negative values. Rather, the pattern repeats after four such cycles, namely, every \(31\) points.

Similarly, the signal \(x[n]=\cos(n/6)\) can be viewed as the set of samples of the signal \(x(t)=\cos(t/6)\) at integer time points. In this case, the values of \(x(t)\) at integer sample points never repeat, as these sample points never span an interval that is an exact multiple of the period, \(12\pi\), of \(x(t)\). Thus, \(x[n]\) is not periodic, although the eye visually interpolates between the sample points, suggesting the envelope \(x(t)\), which is periodic.


Example 1.6

Suppose that we wish to determine the fundamental period of the discrete-time signal


The first exponential on the right-hand side of eq. (1.59) has a fundamental period of 3. While this can be verified from eq. (1.58), there is a simpler way to obtain that answer.

In particular, note that the angle \((2\pi/3)n\) of the first term must be incremented by a multiple of \(2\pi\) for the values of this exponential to begin repeating. We then immediately see that if \(n\) is incremented by 3, the angle will be incremented by a single multiple of \(2\pi\).

With regard to the second term, we see that incrementing the angle \((3\pi/4)n\) by \(2\pi\) would require \(n\) to be incremented by 8/3, which is impossible, since \(n\) is restricted to being an integer. Similarly, incrementing the angle by \(4\pi\) would require a non-integer increment of 16/3 to \(n\). However, incrementing the angle by \(6\pi\) requires an increment of 8 to \(n\), and thus the fundamental period of the second term is 8.

Now, for the entire signal \(x[n]\) to repeat, each of the terms in eq. (1.59) must go through an integer number of its own fundamental period. The smallest increment of \(n\) that accomplishes this is 24. That is, over an interval of 24 points, the first term on the right-hand side of eq. (1.59) will have gone through eight of its fundamental periods, the second term through three of its fundamental periods, and the overall signal \(x[n]\) through exactly one of its fundamental periods.


As in continuous time, it is also of considerable value in discrete-time signal and system analysis to consider sets of harmonically related periodic exponentials-that is, periodic exponentials with a common period \(N\).

From eq. (1.56), we know that these are precisely the signals which are at frequencies which are multiples of \(2\pi/N\). That is,


In the continuous-time case, all of the harmonically related complex exponentials \(e^{jk(2\pi/T)t}\), \(k=0,\pm1,\pm2,\ldots\), are distinct.

However, because of eq. (1.51), this is not the case in discrete time. Specifically,


This implies that there are only \(N\) distinct periodic exponentials in the set given in eq. (1.60). For example,


are all distinct, and any other \(\phi_k[n]\) is identical to one of these (e.g., \(\phi_N[n]=\phi_0[n]\) and \(\phi_{-1}[n]=\phi_{N-1}[n]\)).



The next tutorial discusses about the unit impulse and unit step functions

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