HEWLETT-PACKARD
JOURNAL
T E C H N I C A L
I N F O R M A T I O N
F R O M
T H E
- h p -
Vol. 7 No. 3
L A B O R A T O R I E S
NOVEMBER, 1955
'UBLISHED BY THE HEWLETT-PACKARD COMPANY, 275 PAGE MILL ROAD, PALO ALTO, CALIFORNIA
Square Wave and Pulse Testing
of Linear Systems
A>J arbitrary input wave, when transmitted
through even an ideally linear system, can
be altered in a number of ways. Such a wave
can be altered, for example, in size, shape, and
time of occurrence. There is, however, one class
of waves or functions — the sinusoids — that a
linear system can alter in only two ways. The
response of any linear system to a sine wave is
another sine wave differing from the original
at most in amplitude and phase. This cardinal
fact gives sine waves their unique position in
communication theory and further gives physi
cal significance to Fourier analysis and to the
whole concept of frequency spectra. Since any
input can be represented as the sum of a num-
ber of sinusoids, the output from a linear sys
tem will consist of these same sinusoids modi
fied only in amplitude and phase. Each will be
transmitted by the system as if it alone were
present (superposition). The simple sum of the
output sinusoids will be the output wave.
The way that the system modifies sine waves
of all frequencies (the system amplitude and
phase characteristic) thus constitutes a com
plete description of the system in that it enables
the output to be computed for any input. The
procedure for such a computation can be rep
resented graphically as shown at left.
If not already known, the spectrum of the
input wave is found by evaluating the Fourier
transform (A to B in the diagram). The system
multiplies the input spectrum by the transmis
sion (amplitude and phase) characteristic <£(<")
to give the spectrum of the output (B to C).
The inverse Fourier transform of the output
spectrum is the output time function (C to D).
Although the above method of computing is
an indirect method, it is often actually easier
than the direct method. The direct method in
volves evaluation of the convolution integral
(often called the superposition or duHamel's
integral):
h(t)«ff(T) ^(t-i-)dr. (1)
Diagrammatic representation of steps necessary to calcu
late how a time function is modified by a linear system's fre
quency characteristic. Direct calculation involves fewer steps
but is often more difficult.
P R I N T E D
I N
U S .
<#>(T) is the impulse response of the system.
</>(*-T) is therefore this same function reversed
left to right and displaced by an amount /.
What (1) says is that, to find the output, the
product of the input and this reversed, dis
placed impulse response must be integrated.
The result will be a function of the displace
ment, /. In other words, the input wave must
C O P Y R I G H T
A .
© Copr. 1949-1998 Hewlett-Packard Co.
1 9 5 5
H E W L E T T
-
P A C K A R D
C O .
be scanned with the reversed im
pulse response.
Equation (1) is easy to derive by
considering the successive ordinales
of f (t) to be a succession of impulses
and applying the superposition theo
rem. Equation (1) is also easy to eval
uate on occasion, but more often it is
difficult. By contrast, rather com
plete tables of transforms exist, so
that getting from A to B and from
C to D often involves merely using a
table. The intermediate step from
B to C is accomplished merely by
multiplying the input spectrum F(o>)
by the system's transmission charac
teristic <¿>((o). The situation is analo
gous to the use of logarithms when
raising a number to a power. To
compute 7^, for example, involves
looking up the logarithm of 7(A to
B), multiplying by ir (B to C), and
looking up the antilog for the an
swer (C to D).
Even more impressive is the case
where f(t) and h(t) are known and
the system frequency characteristic
and impulse response are desired.
The direct solution involves solving
(1) as an integral equation. But by
the indirect transform method the
answer is simply
<j>(t) is then the inverse transform
of<£(«>).
The above has shown how a
knowledge of the steady state (sine
wave) characteristic of a system en
ables the output waveform to be
computed for any known input
waveform. Even if the exact input
waveform is not known, however, a
knowledge of the steady state per
formance of the system enables a
picture to be gained of how the sys
tem will transmit the input wave.
All that may be known, for example,
is that the input may contain all fre
quencies over a certain band (e.g.,
speech or music) and that the system
must be able to transmit this whole
class of inputs without distortion.
This requires that the output be a
replica of the input except for pos
sible changes of size and delay;
thatis: h(t) =K f(t-t0)
where K is a constant and t0 is a per
missible delay. In the frequency do
main this requires that the output
spectrum be the input spectrum
modified only by a constant ampli
tude factor K, and a linear phase
shift, </> = — w t0; that is
H (w) - K F((j) 6 i\ ^ z
The system frequency characteristic
must therefore be flat, with linear
phase over the band of frequencies
to be transmitted:
< i ( u j )
-
K
6
1 ^
^
2
RESPONSE OF LINEAR
SYSTEMS TO IMPULSES
While steady state measurements
are very useful for the reasons dis
cussed above, they are also quite
time consuming to make. For many
purposes the transient response of
the system to certain particular types
of input waves may provide all the
information necessary. In fact, if the
input wave is properly chosen, such
a transient measurement provides
exactly the same information as a
steady state measurement but pro
vides it in a different form.
Consider, for example, the case
where the input, i(t), is an impulse
of negligible duration and, say, unit
area. The spectrum of such an im
pulse contains all frequencies. The
frequencies all have the same ampli
tude and are all in phase in the sense
that they all add at t = 0. In other
•words the spectrum of such an im
pulse is a constant F(w) = 1.
Now from the superposition theo
rem, it makes no difference whether
all frequencies are introduced one
after another, as in steady state test
ing, or simultaneously by applying
an impulse. In either case any fre
quency will be modified in the same
way by the linear system. Impulse
testing thus might be said to be
equivalent to an instantaneous
steady state test. They both give the
same information, but the results
must be interpreted in either case.
When the input is an impulse, the
input spectrum is F(«) = 1 and the
output spectrum is H(o>) = 4>M. In
© Copr. 1949-1998 Hewlett-Packard Co.
other words, the response of a linear
network to an impulse is a pulse
whose spectrum is the amplitude and
phase characteristic of the network.
Obviously, the interpretation of
impulse tests involves a familiarity
with the spectra associated with a
wide variety of time functions and
vice versa. A Table of Transforms
published in these pages some time
ago illustrated a number of time func
tions with their associated spectra.
Impulse testing has advantages,
but it also has two severe disadvan
tages. First, the impulse must be
short compared with the duration
of the finest detail of the output
transient which is to be reproduced
accurately. In other words the spec
trum of the impulse must be flat over
the entire frequency range of the
device under test. To get appreciable
response, then, often requires that
the impulse be so large in amplitude
that the device under test is driven
out of its range of linear operation.
The second disadvantage is that,
when testing wide band devices, the
low frequency effects are hard to ob
serve. This is because only an insignif
icant amount of the wide input spec
trum is deleted by the low frequency
cutoff of the device under test.
RESPONSE OF LINEAR SYSTEMS
TO STEP FUNCTIONS
The disadvantages of impulse test
ing are avoided by vising step func
tions. With a step the rise time can
be as short as desired without a re
sulting increase in amplitude. Since
the spectrum of a unit step is F(o>) =
I/jo), more energy is concentrated at
the low end of the spectrum. Low
frequency effects are thus placed on a
more nearly equal footing with high
frequency effects.
A unit step is the integral of a unit
impulse. The step function response
of a system might thus be obtained
in any of the three ways indicated
in the diagram. Part (a) of the dia
gram shows a straightforward test
with a step function generator. In
(b) the step generator is replaced by
*B. M. Oliver. "Table of Important Trans
forms," Hewlett-Packard Journal, Vol. 5, No.
3-4, Nov. -Dec., 1953.
Three possible test arrangements for obtaining a system's step-function response.
a combination of an impulse genera
tor and an integrator. In (c) the in
tegrator has been interchanged with
the system under test. Since both are
linear systems the output is unaf
fected. (c) illustrates the important
fact that the step junction response
of a linear system is the integral of
the impulse response. Further, the
spectrum of the step function re
sponse is 1/jia times the steady state
amplitude and phase characteristic
of the system.
Conversely, the impulse response
is the derivative of the step response.
SQUARE WAVE TESTING
The accompanying table (I) shows
the step function responses for some
typical common networks. If the
condition stated below is met, these
responses will also be the response to
the (positive) step of a square wave,
because a square wave can be consid
ered to be a succession of alternate
positive and negative steps. The
table includes a few explanatory re
marks with each response.
In order for the response to each
step of the square wave to be identi
cal with a system's step function re
sponse, the square wave frequency
must be low enough so that the in
dividual transients do not overlap.
In other words the square wave fre
quency must be less than l/2t, where
t is the time required for the step
response to reach a constant value
within the desired accuracy. Cases
*Assuming either a match at all junctions or
frequency insensitive mismatch.
where long duration square waves
should be used are where sharp ir
regularities exist in the frequency
characteristic. Typical cases of sharp
irregularities are low end cutoffs or
sharp resonances anywhere in the
pass band.
If high-end cutoffs are being ob
served, a relatively high repetition
frequency with its widely-spaced
spectral lines is usually permissible.
The reason for this is that high-end
cutoffs are relatively broad frequency
effects (consume only a short time
in the time domain).
Sharp mid-band effects may be ex
plored even with high repetition
frequencies if the repetition fre
quency is variable. Sweeping the
repetition frequency will always
cause some harmonic of the square
wave to coincide with a sharp midband effect and thus produce an ob
servable transient. In fact, long dur
ation resonances that are scarcely
visible in the step function response
because of their low amplitude will
become quite prominent when the
proper frequency square wave is
vised. The reason for this is that the
proper frequency square wave will
cause the successive excitations of
the resonance to reinforce preceding
excitations with the result that a
much larger amplitude oscillation is
produced.
The complete step function re
sponse of a system having a low fre
quency cutoff is only displayed if the
repetition frequency is much less
than the frequency of the cutoff.
Since it is often inconvenient to use
such a low frequency, it is customary
to use a frequency such that the tran
sients from the successive positive
and negative steps do not vanish
completely but rather overlap con
siderably. In such cases the square
wave response needs to be inter
preted. Table II shows some typical
overlapped low frequency responses.
-B. M. Oliver
TABLE II
EFFECTS OF TYPICAL LOW END DISTORTIONS ON SQUARE WAVE
(A)
LOW FREQUENCY PHASE LEADING
(0
LOW FREQUENCY AMPLITUDE
LOW FREQUENCY PHASE LAGGING
(D)
LOW FREQUENCY AMPLITUDE
(E)
(F)
LOW END SIMPLE RC CUTOFF (ASD)
RESULT OF PHASE COMPENSATING E
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