Digital Signal Processing Reference
In-Depth Information
All these test signals are generated on the basis of formulae by means of a computer and
not by means of special analog circuits. Each of these test signals has certain advantages
and disadvantages which will be outlined briefly here.
The
-pulse
How can the oscillation properties of a car - the mass of the car, the springs, shock
absorbers form a strongly damped mechanical band pass - be measured in a straightfor-
ward way? Easy - by driving over a pothole in the road at speed! If the car continues
vibrating for a longer time - because of the
UP
it is a narrow band system - the shock
absorbers are not in order, that is, the car as a mechanical oscillation system is not suffi-
ciently damped.
δ
The electrical equivalent to the pothole in the road is the
-pulse. The reaction of a system
to this spontaneous, extremely short-lived deflection is the so-called
δ
-pulse response at
the output of the system (see Illustration 108). If it is extended in time the frequency
domain is according to
UP
strongly restricted, i.e. a kind of oscillating circuit occurs. Any
strongly damped oscillating circuit (intact shock absorbers or ohmic resistance) is broad-
band, i.e. it therefore produces a brief pulse response. This is the only way the physical
behaviour of filters (including digital filters) can be understood.
δ
The
-pulse response (generally called the “
pulse response” h(t)
) provides via the
UP
qualitative information on the frequency properties of the system being tested. But it is
only the FT that provides precise information on the frequency domain. It alone shows
what frequencies (and their amplitude and phases) the pulse response contains.
δ
If a system is tested by a
-pulse it is - unlike the sweep signal - tested simultaneously
with all frequencies (sinusoidal oscillations) of the same amplitude. For example at the
output of a high pass filter the low frequencies below the cutoff frequency are almost com-
pletely lacking. The sum of the (high) frequencies allowed to pass from the pulse response
h(t). In order to obtain the frequency response the pulse response h(t) must simply under-
go an FT.
δ
The importance of the
-pulse as a test signal is based on the fact
that the FOURIER transform FT of the pulse response h(t)
already represents the transfer function/frequency response
H
(f)
of the system tested.
δ
Definition of the
transfer function
H
(f): For every frequency, amplitudes and phase
displacements of u
out
and u
in
are compared with each other.
H (f) = ( Û
out
/Û
in
) (0 < f <
∝
)
Δϕ
= (
ϕ
out
-
ϕ
in
) (0 < f <
∝
)
Conclusions and notes:
• Unlike the sweep signal in the case of the
-pulse the system is measured
simultaneously with all frequencies (Û = constant).
δ
• As in the case of the
-pulse Û
in
= constant for all frequencies, the amplitude spectrum
of the pulse response forms the curve of the absolute value H(f) of the transfer
function. On account of this relatedness the pulse response is represented
internationally as h(t).
δ
