1-Waveform

THE SINUSOIDWe start with a very simple waveform, a sinusoid. As shown in Figure   1 (a), we first visualize a steady counterclockwise rotation— perhaps of a light at the end of a crank of length A.

Figure 1

The instantaneous position of the light is given by the phase angle . The light rotates so many times per given time period; this number is the frequency f. Then we plot, on the graph at the right, the vertical position of the light above its start position, as a function of time. After 45° of rotation ( Figure   1 (b)), the light is 0.707A units above its start position. After 90 ( Figure   1 (c)), it is A units. After 135° ( Figure   2 (d)),

Figure 2

it is again 0.707A units, and so on ( Figure   2 , Figure 3 (e -h) .

Figure 3

The end result, after a 360 rotation ( Figure   3 (i)) is a sinusoid.

One full revolution of the crank provides one cycle of the sinusoid. Further revolutions produce further cycles, each indistinguishable from the first.

Figure   4 shows the characteristics of a sinusoid.

Figure 4

The amplitude A corresponds to the length of the crank. The double amplitude 2A is called the peak-to-peak amplitude (or the peak-to-trough amplitude). The period T is the repetition time; clearly this is the reciprocal of the frequency: T = 1/f. If the units of time are seconds, the units of frequency are cycles/second, or hertz (symbol Hz). The shape of a sinusoid is completely specified by its amplitude and its frequency.

Phase

In Figure   5 (a), we see a sinusoid in which the time origin was taken at the instant when the phase angle was zero.

Figure 5

In Figure   5 (b), we see a second sinusoid, of the same frequency and amplitude, occurring one-quarter of a period earlier. The second sinusoid is said to lead the first (or the first is said to lag the second) by a phase angle of 90°. From Figure   5 (c) we see that a phase lead (or lag) of 180° corresponds to a reversal of polarity (that is, a peak instead of a trough, or a trough instead of a peak). A phase lead (or lag) of 360° leaves a sinusoid unchanged.

The wave of Figure   5 (a), in which the phase is such that there is a zero-crossing at the time origin, is a sine wave. The wave of Figure   5 (b), for which the phase is such that there is a peak at the time origin, is a cosine wave; we shall take this as our basic standard waveform.

AMPLITUDE, POWER, AND ENERGYSinusoids are very familiar to us: the cycle of the seasons, the motion of a long pendulum, the ebb and flow of the tides. Our household electricity supply is "alternating current" (ac); the variation of voltage with time is sinusoidal, with a frequency of 50 or 60 Hz. Indeed, household electricity supply is a good example with which to illustrate some fundamental relationships.

If we could "see" the voltage of the household electricity supply, it would look like Figure   1 (a).

Figure 1

For some of the time, near the peaks, the voltage is large; at other times, near the zero-crossings, it is small. This is important when we come to use the electricity — for example, in a heater. The rate at which the heater can deliver heat is its power, measured in watts or kilowatts; this is proportional to the square of the voltage. The square of the voltage is shown in Figure   1 (b); this is again a sinusoid, but at double the frequency. The high values correspond

to the positive and negative peaks of the original voltage sinusoid, while the low values correspond to the zero-crossings. Full power is generated at the high values, but very little at the low values. The power effective for producing heat is clearly the mean value, shown dashed in Figure 1 (b); this is only half the value suggested by the square of the voltage. This leads to the concept of a root-mean-square (rms) voltage, which is the voltage effective for producing heat. It is the square root of half the square of the voltage amplitude, or , or 0.707A. Use of this value in power calculations compensates for the fact that the voltage is fully effective only near its peaks.

When we speak of a 110-volt supply or a 240-volt supply, we mean that the rms voltage is 110 volts or 240 volts; the instantaneous voltage of a 110-volt supply varies sinusoidally between +156 volts and -156 volts.

The energy consumed while a heater is switched on is proportional to power times time; it increases steadily in proportion to 1/2 A2t, and is measured in watt-hours or kilowatt-hours.

It is important that we keep clear in our minds the distinctions between amplitude, power, and energy. We may summarize the amplitude relationships associated with a sinusoid:

amplitude (A):

instantaneous value = A sin = A sin 2(f) (t) = A sin 2 t / T * peak-to-peak amplitude = 2A rms amplitude =

mean power: proportional to (1/2) A2

energy proportional to (1/2) A2t

* If the angle is measured in degrees, we replace 2 by 360°.

Relative Amplitudes and the Decibel Scale

Sometimes we are concerned less with absolute amplitudes than with relative amplitudes. Particularly when the range of amplitudes is large, we express the ratio of the amplitudes using the decibel scale.

The formal relation — dB = 20 log (A2/A1) — is graphed in Figure   1 .

Figure 1

However, several much-used values are worth memorizing:

0 dB amplitudes equal

+1 dB about 10% more

-1 dB about 10% less

+3 dB about 40% more

(approximately )

-3 dB about 30% less

(approximately )

+6 dB about double

-6 dB about half

+10 dB about 3 times

+20 dB 10 times

+40 dB 100 times

+60 dB 1000 times

Thus, if an amplitude is 70.7% of a reference amplitude, it is said to be -3 dB, or "3 dB down." The addition of decibel values corresponds to the multiplication of ratios; thus, from the above memorized values we know immediately that 26 dB (20 + 6) represents an amplitude ratio of 20 (or 10 2), and that 54 dB (or 60 - 6) represents an amplitude ratio of 500 (1000 2).

Addition of Sinusoids of the Same Frequency

In Figure   1 (a), we see two cosine waves of the same frequency and amplitude; they are also in phase.

Figure 1

If we add them together (for example, by adding corresponding ordinates p at closely spaced values of time), we obtain Figure   1 (b). This is another cosine wave, of the same frequency and with double the amplitude; it is in phase with the two constituents.

In Figures 2-5 , we do the same for two waves of the same frequency and amplitude, but with different phase. Always we obtain a sinusoid of the same frequency. The amplitude of the new wave is double if the phase is 360° (or 720° or 1080° ... Figure   2 ); we call this reinforcement, or constructive interference.

Figure 2

The amplitude is zero if the phase is 180° ( Figure   5 ); we call this cancellation, or destructive interference.

Figure 5

Otherwise, the amplitude is intermediate ( Figure 3 ,

Figure 3

Figure 4 )

Figure 4

What happens if we add a large number N of waves, all having the same frequency and amplitude? In Figure   6 we see 36 such waves, all in phase.

Figure 6

Obviously, the sum has 36 times the constituent amplitudes; to keep the summed wave on the paper, we divide it by the number of constituent waves (we normalize it), and emerge with a normalized sum equal to the constituent amplitudes.

In Figure   7 we shift the 36 constituent waves by regular phase increments of 10°, right round the phase clock.

Figure 7

The sum is zero and the normalized sum is zero; for each wave there is another which is systematically 180° out of phase and so cancels it.

In Figure   8 the phase shifts applied to the 36 waves are random.

Figure 8

The sum is still sinusoidal, but its amplitude is only about six times () the constituent amplitudes. The normalized sum is , or one-sixth of the constituent amplitudes; this means that it tends to zero as N is increased.

Since this conclusion is not intuitively obvious, we consider a musical analogy. Figure   9 suggests a number of sound sources equidistant from a listening microphone.

Figure 9

Initially we may suppose that the sources are identical loudspeakers, driven by the same signal from a musical synthesizer. Let us suppose that we set the synthesizer to deliver a sinusoid at 3000 Hz; it would sound like a sustained and fairly high-pitched note on a flute. If we drive only one of the sources, we establish a reference amplitude from the microphone; as we bring in more sources, the amplitude increases in proportion. This is our counterpart of Figure   6 . Each additional source contributes as much amplitude as the first.

Now let us replace the loudspeakers by human flute players with identical flutes, all playing the same note. The difference is that the sound from human players (being derived from the flow of air out of different lungs) has randomly different phase. As before, the first flute player establishes a reference amplitude from the microphone. As more players join in, the sound heard by the listener does increase in amplitude (of course it does) but not by as much as with the synthesizer. Indeed, if we have to pay the musicians, we soon decide that the greatest cost-benefit is obtained from the first, and very little from the last. The cost-effectiveness decreases with every player; this is the counterpart of the normalized sum of Figure   8 .

What happens if we add sinusoids of the same frequency but different amplitude? The simple case of two waves ( Figures 10-13 ) is sufficient to establish the principle. We obtain a new sinusoid, also of the same frequency, whose amplitude again depends on the phase between the waves: at 360º (or 0º) it is the sum of the constituent amplitudes ( Figure   10 ),

Figure 10

at 180º it is the difference ( Figure   13 )

Figure 13

; and at intermediate phases it is intermediate ( Figure 11 ,

Figure 11

Figure 12 )

Figure 12

From all the examples of this section we can infer an important generalization:

The addition of sinusoids of the same frequency always yields a sinusoid of the same frequency.

Addition of Sinusoids of Different Frequency

In Figure   14 we add two sinusoids of the same amplitude but different frequency.

Figure 14

Sometimes the two waves are in phase; the amplitude is then doubled. As the phase gradually changes, the amplitude decreases to zero and then increases again. This is the musical phenomenon of beats; two closely spaced musical tones appear to swell and diminish cyclically. The new waveform looks like a sinusoid contained within (or modulated by) an envelope (dashed in Figure   14 (c)). The frequency of the new wave is between the two constituent frequencies; the sinusoid defining the envelope rises and falls slowly if the constituent frequencies are close, and faster if they are widely different.

Addition of Harmonics, and the Fourier Series

Consider the addition of two sinusoids. When the second sinusoid has a frequency which is 2, 3, 4, ... times the frequency of the first (the fundamental), it is called a harmonic. Thus the third harmonic has a frequency three times that of the fundamental. Following musical terminology, we call the interval between any fundamental and its second harmonic an octave.

In music, the addition of harmonics to a fundamental is of major importance; it accounts for the different sounds from different musical instruments, and it forms the basis for the orchestral blending of instruments. Thus, in Figure   1 we visualize an instrument whose tone is rich in the second harmonic (indeed,

Figure 1

the amplitude of the second harmonic is equal to that of the fundamental), and in Figure   2 an instrument whose tone is rich in the third harmonic.

Figure 2

The two instruments are playing the same note (that is, the same fundamental), but they make a different sound. Instead of emitting a sinusoid (like the "thin" sound of a flute), they emit complex waveforms (such as those shown as sum waveforms) by which we can recognize their distinctive "rich" sounds.

In signal theory, the addition of harmonics to a fundamental is also of major importance. As we look at Figures 1 - 6 ( The synthesis of complicated repetitive waveforms by the Fourier series ), we can see why this is so; we have six complex and distinctive sum waveforms which are synthesized by adding together just two sinusoids. Let us look at the six cases in turn.

In Figure   1 a fundamental and its second harmonic are added, with their peaks at the time origin; they are both cosine waves. The sum waveform repeats at the same frequency as the fundamental, but the presence of the second harmonic is also evident.

In Figure   2 the waves are again cosine waves; as before, the sum waveform repeats at the frequency of the fundamental, but now the third harmonic is evident.

Figure   3 repeats Figure   1 ,

Figure 3

except that the phase of the second harmonic is shifted by 90º. We still see the two component frequencies in the sum waveform, but the shape of the sum waveform is quite different. Evidently a wide variety of sum waveforms can be obtained, from the same sinusoidal components, by changing the phase relation. Further, we note that the peak amplitude of the sum waveform in Figure   3 is less than that in Figure   1 , because the peaks of the constituent sinusoids never occur at the same time.

In Figure   4 both components are sine waves, and the amplitude of the second harmonic is halved.

Figure 4

The sum waveform is noticeably less rich in the higher frequency. Since both components have a zero-crossing at the time origin, so does their sum.

In Figure   5 we double the amplitude of the second harmonic, and impose the phase shifts in the opposite direction.

Figure 5

The sum waveform is now rich in the higher frequency, but the repetition is still at the frequency of the fundamental.

Figure   6 returns to cosine waves of equal amplitude, and illustrates that even if the fundamental is not present among the components it is apparent in the sum.

Figure 6

The frequency of the fundamental is always the highest common factor of the frequency of the components; thus the addition of component waves of frequency 3f and 4f produces a complex waveform which repeats with a frequency of f.

The extreme diversity of the sum waveforms in Figures 1-6 prepare us for a very important general principle:

Any waveform which repeats with frequency f, however complicated it may appear, can be regarded as formed by the addition of sinusoids of frequency f, 2f, 3f, 4f ....

Thus the square wave, the triangular wave, and the heartbeat of Figure   7 — to the extent that each repeats precisely — can be regarded as formed by the addition of sinusoids.

Figure 7

Which could hardly be said to be self-evident. Further, since the fundamental repetition is the same for all three examples, the component sinusoids are the same sinusoids; only their relative proportions and their phases are different.

This remarkable principle is known as the Fourier series. Not surprisingly, Joseph Fourier (1768-1830) had a hard time convincing his contemporaries that it was true. However, one of his friends was Napoleon; perhaps that helped.

Since we have agreed that a sinusoid is completely specified by its frequency, its amplitude, and its phase, we can now restate the above principle:

Any repetitive waveform can be regarded as completely specified by the amplitude and phase of the fundamental and each harmonic. The fundamental and its harmonics are called Fourier components.

Line Spectra

We can easily represent on a diagram the amplitudes and phases of the Fourier components. Figure 1 ,

Figure 1

Figure 2 ,

Figure 2

Figure 3 ,

Figure 3

Figure 4 ,

Figure 4

Figure 5 , and Figure 6 show a series of sinusoids and their sums.

Figure 5

Figure 7

Figure 7

, Figure 8

Figure 8

, Figure 9

Figure 9

, Figure 10

Figure 10

, Figure 11

Figure 11

, and Figure 12 depict the corresponding amplitudes and phases as a function of frequency.

Figure 6

Each situation needs two diagrams, which we call an amplitude spectrum and a phase spectrum. The amplitude spectrum displays the amplitude of the fundamental and of each harmonic as a heavy vertical line at the discrete frequency of each component.

Figure 12

The phase spectrum gives the phase angle of each component at the time origin. In this illustration, we have taken cosine waves, each with a peak at the origin, as our reference.

Any repetitive waveform is completely specified by its amplitude and phase spectra. Conversely, any pair of amplitude and phase spectra completely specify a unique repetitive waveform.

For the case where our waveforms represent variations as a function of time (for example, in a cardiogram, along seismic traces, or in stock-market charting), we now see an interchangeability between a waveform as a function of time and its two spectra as functions of frequency; we see a reciprocal correspondence between measurements in the time domain and measurements in the frequency domain. If our waveforms represent variations as a function of space (for example, in a gravity survey, in medical tomography, or in pattern recognition), we see the same duality between the spatial domain and the spatial-frequency domain. In each case, the bridge between the two domains is provided by Fourier. Sometimes we may choose to specify our waveforms in one domain, sometimes in the other; we have freedom of choice, because we can always use the Fourier concept to transform from one domain to the other.

The Fourier Integral

In looking at Figure   1 , we may note that the sum waveform repeats with a frequency which is the highest common factor of the component frequencies.

Figure 1

Figure 2 ,

Figure 2

Figure 3 ,

Figure 3

Figure 4 , and Figure 5 explore this further.

Figure 4

In Figure 2 ,

Figure 5

and Figure 3 we see three cosine waves, of frequency 4, 6, and 8 Hz; they are in phase at the time origin. We see also their sum; it repeats twice per second, because the highest common factor of 4, 6, and 8 is 2. In Figure   4 , and Figure 5 we add components at 5 and 7 Hz, while preserving the zero phase spectrum; the highest common factor is 1, and the sum waveform repeats once per second.

Extending the process, we see that if we were to add components every 1/1000 Hz between 4 and 8 Hz, the resulting sum waveform would repeat only after 1000 seconds. Thus we are led to the concept of the Fourier integral: if the components occur at an infinitesimal frequency spacing, the sum waveform repeats only after infinite time. This releases us from considering only repetitive waveforms; we can now consider transient waveforms, those that occur only once.

Any transient waveform can be regarded as the addition of sinusoidal components which are harmonics of an infinitesimal fundamental frequency.

The importance of this is that we can now apply the concepts of Fourier to pulses — transient signals that do not repeat.

Thus the line spectrum of Figure   4 ,and Figure 5 from 4 to 8 Hz, corresponds to a repetitive sum waveform. The continuous spectrum of Figure   6 ( the counterpart to

fig.3 for a continuous amplitude spectrum ),

Figure 6

which covers the same bandwidth from 4 to 8 Hz, also has a corresponding sum waveform; near the time origin, this sum waveform looks quite like that in Figure   4 ,and Figure 5 but it never repeats.

The spectra of repetitive waveforms are line spectra; those of transient waveforms are continuous spectra.

Line spectra, then, are appropriate to signals like the vibration of an engine, or for assessing the "tone" of a cello. Continuous spectra are appropriate to thunderclaps, percussion instruments, and seismic pulses. And if we find that a heartbeat is not precisely regular, we are not prevented from representing it in the frequency domain; we can take one single heartbeat, and represent it by continuous spectra.

The Continuous Amplitude Spectrum

When we consider line spectra, we are able to draw the Fourier components, and add them to obtain the sum waveform; Fourier's message is evident graphically. How are we to do the same with a continuous spectrum, such as that of Figure   1 (a) ( the concept of spectrum density )? Certainly we cannot draw an infinite number of

sinusoids of infinitesimally increasing frequency.

Figure 1

Further, we can surmise that the total energy in the time waveform (which is finite) should be the same as the sum of the energy in the component sinusoids; this means that it there are an infinite number of components the amplitude of each must be infinitesimally small — which again would be difficult to draw. However, there is a solution.

The solution is based on the observation that we need an infinite number of frequency components only if the time waveform is infinitely long. But if we know that the waveform is half a second long, no ambiguity is introduced if we consider it to repeat every half second; then we know it can be represented by frequency components at intervals of 2 Hz, as in Figure   2 and Figure 3 .

Figure 2

If we know it to be 1 second long,

Figure 3

then the frequency components must be at intervals of 1 Hz, as in Figure   4 and Figure 5 .

Figure 4

Figure 5

Therefore, we can realize the Fourier integral graphically by dividing the continuous spectrum of Figure   1 (a) into a number of equal slices ( Figure 1 (b)), each of which is drawn as one component sinusoid. The frequency of each component is the center frequency of the corresponding slice. The amplitude of each component is the area of the slice — the height times the width. (The vertical axis of a continuous spectrum is therefore properly labeled amplitude spectrum density — though in practice we usually still call it an amplitude spectrum.) And the width of each slice must not exceed the reciprocal of the duration of the transient waveform.

Let us take a case in point. One method of testing the "acoustics" of an auditorium is to fire a pistol at the position of the performer, and to record the resulting transient sound heard at various places in the audience. The sound includes the initial sharp crack, followed by a train of multiple echoes — mostly from the walls and ceiling. In principle, of course, this train goes on forever. As such, it can be truly represented by continuous spectra, in which the sinusoidal components (we say the frequencies") are infinitely long, infinitesimally spaced in frequency, and extend to infinite frequency. But in practice we cannot record waveforms forever; we truncate the train at a point where we think it has fallen to "nothing."

Let us suppose this is five seconds. If we were to form a repetitive signal by repeating this waveform every 5 s, we know that the corresponding spectra would be formed of lines 1/5 Hz apart — with no values between the lines. Similarly, if we consider just the single 5-s segment, we know that the corresponding spectra are now continuous, but these spectra are rigorously defined only every 1/5 Hz. There is therefore an

uncertainty about the spectra of a transient signal that has only finite length. If we had truncated the auditorium waveform at 10 s instead of 5, the spectra would have been better defined (at every 1/10 Hz instead of every 1/5 Hz). And, obviously, a pair of spectra defined only every 1/5 Hz can be reconstituted to a time waveform only 5 s long; after that it must repeat, and we can never recover what has been cut off.

In practice, the one-to-one correspondence of a transient waveform and its continuous spectra is valid only if we are careful about truncation.

With that said, we can continue with our simple graphical representation of the Fourier integral. Figure   6 shows a rectangular amplitude spectrum from 10 to 80 Hz, with all frequencies at the same amplitude ( rectangular spectra such as these are often called "boxcar" spectra ); let us say that we wish to visualize this as a time waveform.

Figure 6

As before, we use cosine waves, and for the present we maintain a zero phase spectrum. We assume that we are interested only in the part of the time waveform near zero time; therefore we allow the waveform to repeat every 0.5 s by taking spectral slices of width 2 Hz. Figure7 shows the Fourier components representing the given bandwidth from 10 to 80 Hz,

Figure7

and Figure   8 shows their normalized sum, this sum repeats every 0.5s.

Figure 8

Figure   8 is therefore the time-domain counterpart of the amplitude spectrum of Figure 6 , for a zero phase spectrum.

The time waveform of Figure   8 is a pulse (or a wavelet), having its maximum amplitude at time zero and tailing away to low values at ±0.25 s (which is as far as we can go). The pulse is, of course, symmetrical; this is a consequence of the zero phase spectrum. The high amplitude at time zero occurs because all the components are in phase. The low amplitudes well away from time zero occur because the different component frequencies produce different component phases, which in turn tend to produce cancellation.

It is worth taking a moment to look at Figure   7 in detail (perhaps around 0.15 s in the middle of the frequency range), and to see how this cancellation occurs. The contribution of any one component is systematically annulled by the adjacent components above and below — at higher and lower frequency.

This observation is very important, and we shall use it many times. Cancellation of one frequency requires the smooth change of phase across that frequency, from lower frequencies to higher frequencies. Naturally, this leads us to see what happens at the abrupt edges of the amplitude spectrum, where — according to our observation — the cancellation cannot be complete. And if we look at the time waveform of Figure   8 , we can see the effect by eye: because 10 Hz and 80 Hz are imperfectly canceled, the tails (or sidelobes) of the pulse appear as an 80-Hz

component riding on a 10-Hz component. By looking at the time waveform, we could estimate the frequencies of the sharp edges of its spectrum.

Sharp edges to an amplitude spectrum prevent the sidelobes of a pulse from dying quickly toward zero.

Figure   9 ,

Figure 9

Figure 10 ,

Figure 10

and Figure 11 maintain the 80-Hz upper limit of Figures 6-8 , but changes the low-frequency limit to 20 Hz.

Figure 11

Figure   12 ,

Figure 12

Figure 13 ,

Figure 13

and Figure 14 modify this again, by cutting the high-frequency limit to 40 Hz.

Figure 14

The summed waveforms at the bottoms of the three figures, therefore, represent 10-80 Hz (which is three octaves of bandwidth), 20-80 Hz (two octaves) and 20-40 Hz (one octave). In each case, some relic of the edge frequencies can be discerned in the sidelobes each side of the pulse.

Looking at these figures again, we find it easy to accept another conclusion:

Short pulses must have large bandwidth.

This is particularly reasonable when we remember that the signal with infinitesimal bandwidth — the sinusoid — goes on forever.

The figures also suggest that we must distinguish two types of bandwidth: bandwidth in octaves and bandwidth in hertz. It is clear that the pulses corresponding to bandwidths of 20-80 Hz and 10-40 Hz (both of which are two octaves) have the same shape, though of course they are differently scaled in time.

For rectangular amplitude spectra and zero phase, therefore, we can say:

Pulses of the same bandwidth in octaves have the same basic shape.

Figure   11 shows pulses for 10-40 Hz and 50-80 Hz (both of which have a bandwidth of 30 Hz), and superimposes their envelopes, which are identical. Again for rectangular amplitude spectra and zero phase, we can say:

Pulses of the same bandwidth in hertz have the same envelope.

According to Figure   15 , we can visualize each pulse as a cosine wave whose amplitude is modulated to remain within an envelope; we are reminded of the phenomenon of beats.

Figure 15

The frequency of the cosine wave is the arithmetical center frequency of the rectangular bandwidth. The central high-amplitude part of the pulse is contained between the first zero-crossings of the envelope, plus and minus; these occur at times equal to the reciprocal of the bandwidth in hertz. If we seek some way of estimating the "duration" of such a pulse, this gives us the key; the central high-amplitude part has a duration of twice the reciprocal of the bandwidth. The highest amplitudes of all (which carry most of the energy) tend to be in the central half of this duration; this illustrates a general and useful conclusion:

The effective duration (in seconds) of a pulse cannot be less than the reciprocal of the bandwidth in hertz.

For many purposes in science we seek pulses that are short and sharp; this is particularly so in echo-ranging, where the object is to measure the time of an echo pulse. We begin to see that "short and sharp" must mean spectra of large bandwidth — both in hertz and in octaves. Further, we often need to minimize the ambiguity represented by the negative excursions of the pulse each side of the central peak (the first sidelobes).

To see what this involves, we look again at Figure 12 , Figure 13 , and Figure 14 . Keeping the upper limit of the bandwidth at 40 Hz, we take the lower limit down progressively from 20 Hz to 0 Hz. The corresponding pulses show smaller and smaller negative sidelobes, down to a minimum of 13% of the central maximum when the bandwidth extends to zero frequency. At this point, of course, we have infinite bandwidth in octaves. Therefore we can say:

The major effect of extending the bandwidth downward is to increase the bandwidth in octaves; this reduces the amplitude of the troughs each side of the central peak, and so improves the "standout" of the pulse.

Figures 2-10 illustrate what happens at the other end:

The major effect of extending the bandwidth upward is to increase the bandwidth in hertz; this increases the center frequency, and so improves the "sharpness" of the pulse.

Before we leave Figures 6-13, we should note that all of the summed waveforms have zero mean (that is, the sum of all the positive and negative excursions is zero), other cases may have bandwiths extending down to zero frequency. Zero frequency is sometimes called "dc " (from direct current, in contrast to alternating current).

Any waveform which does not have zero mean must have some spectral content at zero frequency.

When we are indeed seeking pulses that are short, sharp, and unambiguous, oscillations in the sidelobes of the pulses are unwelcome. These tails result from incomplete cancellation of the components at the sharp edges of the spectrum. One solution, obviously, is to take the lower edge to zero frequency and the upper edge to infinite frequency; then, as shown in Figure   16 , we obtain a spike.

Figure 16

We think of the spike as having infinite amplitude but infinitesimal duration, so that its energy is finite; we shall find that it is a very important waveform indeed.

A spike has a uniform amplitude spectrum extending from zero frequency to infinity, and a zero phase spectrum.

By analogy with light, a uniform spectrum containing all frequencies is sometimes described as "white." In practice, a waveform is effectively a spike if its duration is small relative to that of other waveforms in the context (and, accordingly, if its spectrum remains white up to frequencies higher than others in the context). In the world of radar, a spike must be very short indeed; in the world of the elephant, a friendly nudge would do.

There is another way to be rid of the pulse sidelobes. We see from Figure   17 ,

Figure 17

Figure 18 ,

Figure 18

and Figure 19 that these sidelobes are due to uncanceled edge components, discernible by eye in the summed pulse.

Figure 19

If the problem occurs because of the sharp edges of the spectrum, the answer is obvious: round the corners.

Figure   20 ,

Figure 20

Figure 21 ,

Figure 21

and Figure 22

Figure 22

are the counterparts of Figures 17-19 for an amplitude spectrum with rounded corners; Figure   23 ( the amplitude spectra and pulses are those from Figures 17-22.

Figure 23

The pulses are normalized to the same peak amplitude; without such normalization, the upper pulse would have a larger peak amplitude ) juxtaposes the spectra and the pulses from both figures. They illustrate a general principle which we shall meet many times again:

Pulse sidelobes consequent on sharp corners to the amplitude spectrum can be reduced by rounding the corners.

This advantage is not obtained without sacrifice. As we can see from Figure   23 , the rounding of the corners of the amplitude spectrum reduces the effective bandwidth. Therefore the duration of the central part of the pulse (and in particular the apparent "period" between the two major troughs) is extended.

In terms of Figure   15 this is an envelope effect, remaining true even if the center frequency (that of the modulated cosine wave) is unchanged by the rounding; if the rounding is different at the two ends of the spectrum there may also be a change in the effective center frequency.

If our application requires a short sharp pulse, we have to make a compromise between short effective duration and low sidelobes; this compromise is expressed in the degree of rounding of the amplitude spectrum.

We see that we cannot escape from our previous conclusion that short pulses must have large bandwidth.

A moment ago, we spoke of the apparent "period" between two troughs (or peaks) of a pulse. We recall that the strict term period is defined only for a sinusoid. Only in a loose sense, therefore, can we speak of the period of a pulse, which is formed of many constituent sinusoids. When we do so, we shall keep the word in quotation marks ("period") to remind ourselves of the loose usage.

In the natural world we seldom encounter rectangular amplitude spectra, even with rounded corners. We often encounter spectra of the type of Figure   24 ,

Figure 24

Figure 25 ,

Figure 25

and Figure 26 — with a peak frequency, a low-frequency droop, and a high-frequency droop.

Figure 26

Such a smooth spectrum yields a pulse with much reduced sidelobes; now our pulse is beginning to look more like a heartbeat, a seismic pulse, or an ultrasonic echo.

Intuitively we feel that the very low frequencies and the very high frequencies must be making a negligible contribution to the summed waveform; below what level can we ignore them? Figure   27 ,

Figure 27

Figure 28 ,

Figure 28

and Figure 29 show the effect on the pulse and its sidelobes if we ignore the components whose amplitudes are less than 10% (-20 dB) of the spectral peak,

Figure 29

and Figure   30 ,

Figure 30

Figure 31 , and Figure 32 do the same for a level of 50% (-6 dB).

Figure 31

Where our primary concern is only with the inner high-amplitude part of the pulse, we might accept such approximations, and thus save ourselves some computation.

Figure 32

We must not do this, however, if we are contemplating any later processing of the waveforms to enhance their frequency content at low and high frequencies.

Figure 33 (a) and (b) represent two parts of the spectrum of Figure   30 ,

Figure 33

split at 40 Hz. Let us suppose that we synthesize, separately, the time waveforms corresponding to these two amplitude spectra. They will both have very pronounced sidelobes, "ringing" just below 40 Hz for Figure33 (a) and just above 40 Hz for Figure   33 (b); this, of course, is because of the very sharp spectral corner formed by the split. If we were to add the two time waveforms, would these excessive sidelobes disappear?

All the amplitude spectra we have seen so far have been plotted on linear scales of amplitude and of frequency. For some purposes, we prefer semi logarithmic or logarithmic plots. Figure   34

Figure 34

reproduces the spectrum of Figure   24 ; Figure   35 modifies the amplitude scale to be logarithmic,

Figure 35

and Figure   36 modifies both scales to be logarithmic.

Figure 36

The logarithmic amplitude scale allows us to see the small spectral components, and allows calibration of the amplitude axis linearly in dB. The logarithmic frequency scale expands the lower frequencies, and facilitates the recognition of octaves.

Many natural waveforms have spectra whose limiting slopes represent some power of frequency. Thus in Figure   36 the low-frequency slope is easily measured between 10 and 20 Hz as +18 dB/octave; as the frequency doubles, the amplitude increases by 8 times (23). We know, then, that the amplitude is varying as the third power of frequency. We often express slopes as so many dB/octave, and we should be comfortable with this.

The Continuous Phase Spectrum

In discussing the continuous amplitude spectrum, we assumed that our Fourier components were cosine waves having a peak at the time origin t = 0. This we call zero phase. If all components have a peak at t = 0, their sum is a maximum at t = 0. Further, the symmetry of all components about t = 0 makes their sum symmetrical also. Further again, the rapid development of cancellation between components away from t = 0 has the effect of concentrating most of the energy in the summed waveform into the smallest pulse duration conceivable for those components.

If we use cosine components, a zero phase spectrum produces a symmetrical pulse having the maximum amplitude and minimum duration allowed by the amplitude spectrum.

Figure 1 ,

Figure 1

Figure 2 , and Figure 3 are examples of a zero phase spectrum.

Figure 2

The first change we might consider is a phase spectrum which is +360° or -360° at all frequencies; then each cosine wave is unchanged, and the summed waveform is unchanged.

Figure 3

In Figure 4 we show the phase spectrum to the left, and the summed waveform to the right, all for the amplitude spectrum of Figures 1-3.

Figure 4

Pulse 3 then represents the zero phase spectrum, and pulses 1 and 5 represent the ±360° phase spectra.

Another easy change to consider is a phase spectrum of +180° at all frequencies. Since a phase shift of 180° merely inverts a sinusoid, all the components are inverted, and so are their sums (pulses 2 and 4).

Figure 5 ,

Figure 5

Figure 6 ,

Figure 6

and Figure 7 are the counterparts of Figures 1-3 for a phase spectrum of -90° at all frequencies.

Figure 7

Each cosine wave becomes a sine wave, with a zero-crossing at t = 0. The sum waveform is now skew-symmetric about t = 0. Its peak and trough closest to the time origin are now equal, but of course smaller than the maximum peak of the zero-phase condition. Figure 8

Figure 8

and Figure 9 (developments of Figure 4 for increments of 45° in the constant phase spectra )

Figure 9

repeat Figure 4 , and adds the skew-symmetric pulse ( Figure 9 (k)). The pulse for a phase spectrum of +90° (being 180° out of phase with -90°) is an inverted version, as shown in Figure 8 (g). The same shapes give us ±270° ( Figure 8 (c) and Figure 9 (p)). We often speak of these pulses as "having a 90° shape."

Finally, Figures 8 and 9 add the 45° shapes. These are, as we would expect, intermediate between the 0° shape and the 90° shape. We now see our basic pulse (from Figure 1 , Figure 2 ,and Figure 3 ) rotated one whole cycle, in each direction, around the phase clock. The effect of the different phase spectra is to change the pulse shape cyclically, from symmetry (0º) through skew-symmetry (90°) through inverted symmetry (180°) through inverted skew-symmetry (270°) back to symmetry. The effect of a change of phase sign is to reverse the pulse in time.

One other thing we should notice: the envelope of all the waveforms in Figure 8 and Figure 9 remain centered on t = 0. In this sense the pulse as a whole does not lead or lag the zero-phase version in time.

Phase Spectra- Linear and Curved

Let us consider what happens if a sinusoid’s phase is some variable function of frequency. The first case to consider is the linear phase spectrum of Figure   1 ( the effect of the slope and the intercept, for linear pahse spectra, on the synthesized

pulse of Figures 2-4 )(a).

Figure 1

Each component is now shifted in phase by an amount which is proportional to frequency — a small fraction of a long cycle at low frequencies, a large fraction of a short cycle at high frequencies. The linear relation between phase shift and frequency means that all components are moved in time by the same amount; therefore Figure   2 ,

Figure 2

Figure 3 ,

Figure 3

and Figure 4

Figure 4

become Figure   5 ,

Figure 5

Figure 6 , and Figure 7 .

Figure 6

All the components, and their sum, are shifted in time by an amount which represents the slope of the phase spectrum.

Figure 7

A linear phase spectrum which intercepts the phase axis at 0° does not change the shape of the summed waveform, but moves it in time by an amount equal to the slope of the phase spectrum. The effect is a simple advance or delay in time.

In Figure   1 , therefore, pulse (c) represents the effect of a zero phase spectrum, (f) represents a linear phase spectrum whose negative slope implies a pulse delay of about 11 ms, and (a) represents a smaller positive slope and its corresponding advance of about 8 ms. Figure   1 (d) repeats the same slope as (a), but the intercept on the phase axis is now -360°; this changes nothing.

In applications where our sole concern is with pulse shape, we sometimes loosely speak of pulses (a), (d), and (f) as being zero phase; if we are also concerned with pulse time, then obviously we must keep the distinction between true zero-phase pulses and linear-phase pulses.

What happens it we maintain the linear phase spectrum but change the intercept? Figure   1 (b) shows that if the intercept is ±180° (or ±540°, ±900°, ...), the pulse is simply an inverted version of the one having the same slope and an intercept of 0°. It the intercept is +90° (or ±450°, -270°, ...) the pulse is skew-symmetric, but shifted in time by the slope of the spectrum ( Figure   1 (e)).

The effect of a linear phase spectrum depends on the intercept on the phase axis, and on the slope. The intercept changes the pulse shape from symmetry through skew-symmetry through inverted symmetry through inverted skew-symmetry back to symmetry. The slope advances or delays the envelope of the pulse in time.

With that as background, we are ready to consider arbitrary phase spectra. Obviously, for any given amplitude spectrum there is an infinity of conceivable phase spectra, and hence an infinity of summed waveforms. However, if the phase spectrum is fairly simple we can still preserve our ability to visualize a waveform from its spectra.

Figure 8 ,

Figure 8

Figure 9 ,

Figure 9

and Figure 10 are the counterparts of Figure   5 ,

Figure 10

Figure 6 , and Figure 7 for the curved phase spectrum depicted. We make the judgment that the shape of the pulse will be most affected by the frequencies near the peak of the amplitude spectrum, and we look at the phase spectrum in this region. It is not far from linear, with a slope of about 360° in 60 Hz, which is about 17 ms. Therefore we expect the maximum of the envelope of the pulse to occur at a time of about 17 ms. Further, the backward projection of the linear portion of the phase curve has an intercept of about 180° on the phase axis; we therefore expect the pulse to approximate an inverted version of our reference zero-phase pulse. The actual summed waveform at the bottom of Figures 8-10 therefore makes the pessimists among us feel a little lucky, the optimists a little smug.

If we look closely at the summed waveform, we note that the front of the pulse has a higher-frequency appearance than the tail. Loosely, we might say that the higher frequencies appear to come in before the lower frequencies. At first, this might surprise us, since the phase spectrum tells us that the low frequencies are leading and the high frequencies are lagging. We can see the explanation when we look at the individual components; the major contribution to the summed waveform, at any time, comes from the higher-amplitude components approximately in phase at that time. Early in the pulse these are the higher-frequency components; late in the pulse they are the lower-frequency components. Looking again at the phase spectrum, we see that this is because its (negative) slope is larger at low frequencies than at high frequencies.

The position of any individual component along the time axis is given by its phase angle, which is evident from the phase spectrum. However, the time at which this component and its neighbors make their greatest contribution to the summed waveform is dictated not by the phase angle but by the slope of the phase spectrum.

By this stage, we are ready to accept that simple waveforms (like the click of a switch) have simple phase spectra, while complicated waveforms (like rumbling thunder) have complicated phase spectra. The phase spectrum of Beethoven's Fifth would boggle the mind.

Phase Spectra (Minimum Phase)

The simple phase spectrum used for Figure   1 ,

Figure 1

Figure 2 , and Figure 3 is particular in two ways.

Figure 2

First, it is chosen to be physically realizable — meaning that it yields a waveform which is zero for all times before t = 0.

Figure 3

(Negative times are no problem if we have our waveforms recorded. In real time, however, the phase spectrum of a sneeze must yield complete cancellation of all components before the sneeze occurs; probably we ourselves did not know that the sneeze was coming, but Fourier did.) Second, it is minimum phase. This property is very important in signal theory, and we should explore it further.

An infinite number of phase spectra can be coupled with any given amplitude spectrum to yield an infinite number of summed waveforms. However, many natural waveforms are characterized by a minimum-phase spectrum, which is determined uniquely from the amplitude spectrum.

For any given amplitude spectrum there is only one minimum-phase spectrum; the two together yield a minimum-phase waveform.

The actual calculation of the minimum-phase spectrum from the amplitude spectrum need not concern us here; a graphical treatment of it is given in the section titled "Minimum Phase Property," which appears in this topic of Signal Theory under the heading "References and Additional Information." But we should recognize some of the characteristics of the minimum-phase property:

· The net phase lag across the bandwidth is the minimum possible for a physically realizable waveform;

· Consequently, the (negative) slope of the phase spectrum is also minimum; the minimum-phase property is therefore sometimes called minimum-delay. The energy in a minimum-phase pulse builds up as quickly as the amplitude spectrum allows, and more quickly than for any other phase spectrum; such pulses are sometimes said to be "front-loaded";

· This does not guarantee that the first half-cycle (or leg) of the pulse is the largest. It all depends on the amplitude spectrum, and in particular on the bandwidth. Thus, Figure   4 ( two octave bandwidth ),

Figure 4

Figure 5 ( one octave ),

Figure 5

and Figure 6 ( one-half octave ) which shows minimum-phase pulses for a selection of bandwidths, illustrates that the energy builds up more slowly for a narrow amplitude spectrum;

Figure 6

· Where the amplitude spectrum is simple, with slopes which remain fairly constant over broad ranges of frequency, the minimum-phase angle at any frequency is determined primarily by the slope of the amplitude spectrum at that frequency; the relation is +90° of phase for each + 6 dB/octave of slope.

Figure   7 , then, shows four waveforms which have the same amplitude spectrum but different phase spectra.

Figure 7

Figure   7 (a) is zero phase, which we recognize by the symmetry about the maximum at t = 0; it is not physically realizable. Figure   7 (b) is linear phase, giving a pure delay equal to the negative slope of the phase spectrum; if the sidelobes fall substantially to zero within this delay period, the waveform is approximately realizable. Figure   7 (c) is minimum phase, and Figure   7 (d) is one of the infinite family of non-minimum phase waveforms; both are physically realizable.

Given an amplitude spectrum and a phase spectrum, and using all the rules given above, we can now form in our mind's eye some crude estimate of the corresponding waveform. Conversely, we can look at a waveform and make some crude judgements about its spectra. For the pulse of Figure   8 (a),

Figure 8

for example, we would expect an amplitude spectrum which is broad (because the pulse is short), and which has no sharp shoulders (because the pulse dies quickly toward zero); we would expect a phase spectrum which might be minimum phase but which is certainly not far from linear (because the pulse is nearly symmetrical about its maximum amplitude). Figure   8 (b) must represent a narrow amplitude spectrum, again having no sharp shoulders. Since there is no tendency for either high or low frequencies to dominate either the early or the late part of the waveform, the phase spectrum is not far from linear. Since the pulse is approximately skew-symmetric, negative-going at the center of the symmetry, the intercept of the phase spectrum at zero frequency is about -90°. Figure   8 (c) must represent a fairly broad amplitude spectrum; the phase spectrum is complicated, with a progressively increasing negative slope. Figure   8 (d) must represent a spectrum which has a sharp shoulder, and response to zero frequency.