Download Determination of the LockIn Filter Bandwidth

Determination of the LockIn Filter Bandwidth
In principle, it is simple to measure the power transfer characteristic of a bandpass filter :
You apply “white noise” to the input of the filter and measure the output power spectrum, e.g.
with a spectrum analyzer indicating the spectral power density, S(ω), as a function of frequency.
In practice, however, the task is not so easy, especially not with our Virtual LockIn.
First, there is generally no “white noise” available. So you have to know the input power
spectrum of your “noise”, and you must critically compare the output vs. input spectra.
Second, with our Virtual LockIn, even this procedure is not feasible due to the lack of a analog
input and output of the instrument …
However, in our case the input “noise” can be simulated in the following way :
In white noise, the power density spectrum, N0, is “thought to be” constant over the whole
frequency range, 0 …∞ Hz. Physically, this assumption does make no sense, due to its energetic
divergence to infinity.
As was already discussed in the Chapter Noise Reduction by Narrow Band Filters, a more useful,
but also “theoretical”, approach is the introduction of the “band limited noise”. Here we assume
a - more realistic - constant power density from ω1 … ω2 < ∞, say Nin = 1 W/Hz = “Unity Power
Density” (UPD), for example. This UPD may be caused by a unity rms voltage drop across a
resistance of 1 Ω.
In our LockIn experiment, we simulate this unity rms noise voltage : Instead of taking a contiuous
voltage spectrum of 1 V/Hz, we take a spectrum of discrete frequencies, say 1, or 10, or … Hz
apart, at our convenience, with unity voltage.
Evidently, the spectral response of the LockIn filter will be the same, irrespective of the origin of
the input signals, be they a white noise with 1 Vrms spectral voltage density, or be they a discrete
frequency “set” of 1 Vrms each.
The basic advantage, however, in our case is that we have a discrete input frequency spectrum
available in Part Two, the response of which may be calculated by the “familiar” correlation
function.
For input frequencies ω = ω1 … ω0 = ωref … ω2, the correlation function(s), eqs. (6, 6a), now
becomes
1
R(nT0 , ω ) =
T
with δ = 0.
T
∫
0
1
f (ω0t ) ⋅ f (ωt ) dt =
nT0
nT0
∫ sin(ω t ) ⋅ sin(ωt ) dt
0
0
As we have seen in eqs.(13, 13a) , this expression represents the Fourier transform, F(ω), or
spectral value of a switched sine wave, sin(ω0t), at the frequency ω.
By plotting these values over the interesting frequency range, determined in priciple by the
bandwidth of the LockIn filter, as it is done in the V(f) plot of Part Two for the corresponding
frequency ratio ω/ω0 = f/f0, we get the complete response function of our LockIn with respect to
the frequency content of the input signal, depending on the time constant, nT0.
Now, the interesting power density response, S(ω), can simply be obtained by applying eq. (18),
i.e. by squaring the F(ω) values obtained in our V(f) plots, i.e. Sout (ω ) = F (ω )
2
for unity
input power, or voltage, resp., density.
Exactly this will be done by clicking the “Power Density Spectrum” button in the V(f)
window. The output power density spectrum will be displayed in relative units. This helps further
to determine the “real” bandwidth of the LockIn filter :
∞
The total output power is obtained by integration,
or, in our case, by summation,
Pout =
1
p = Pout =
Sout ( ω ) d ω = N out ⋅ B ,
2π ∫0
1 n
2
F (ωi ) = 1 ⋅ B
∑
n i =1
Here B is the bandwidth, Nout, the maximum output power density is “normalized” to unity, as is
the maximum discrete power level at the reference frequency ω0, p(ω0).
Exactly this procedure yields the ( green ) Output Power Density diagrams, in units relative to
p(ω0), depicted in the V(f) plot of Part Two, the “Power Density Spectrum” button being
activated.
The sum of the complete green area (i.e. the total output power ) can be imagined as the
rectangle around ω = ω0, or f/f0 = 1, respectively, with height 1, the normalized maximum output
power density, times bandwidth B.
The bandwidth calculated with this procedure is the width of the area depicted in clear green in
the Power Density Spectrum plot.
As you will see, this width corresponds – within the inherent experimental, fit, and mathematical
interpolation errors - to the half width of the “main lobe” of the output power spectrum, exactly
as in the case of “normal” bandfilters in communication, or technical applications.
So, despite the absence of a physical bandpass filter, with a low pass filter only, the LockIn
behaves as a true band pass filter, its bandwidth depending on the time constant of the low
pass filter. By augmenting the time constant, you may in fact achieve “theoretical” band widths in
the range of f0*10-8 Hz, or filter “quality factors” of Q = 108, respectively.
This, in fact, is the true “mechanism” of improving the S/N ratio of signals applied to the
LockIn.