Download Sampling Theorem

Sampling Theorem,
frequency resolution &
Aliasing
The Sampling Theorem will be the single
most important constraint you'll learn in
computer-aided instrumentation design
Consider a periodic function, with period T:
Fourier Series
Where we have applied a low-pass “anti-aliasing” filter
to the raw analog signal ending up with h(t), a signal
with a highest freq component < 10*ω0
Ak , Bk: Fourier spectrum coefficients
Or…generate 20 independent equations by sampling. We must
sample 20 times during one period. (If we sample over more than
one period, we may not end up with independent equations.) Each
equation will be of the form: {tm} is the set of sample times…
The matrix (C) of coefficients looks like:
Let the 20 x 20 matrix be C:
with the
solution
Where matrix C must have a non-zero determinant
Coefficients become spectrum: magnitude
Phase change as a function of ω:
Spacing of samples
If the sample spacing is uniform there will be 2N segments of
time, (time between samples) Therefore the sample rate is
2N/T.
The maximum frequency in h(t) is
Therefore you must sample at at least
2*fmax to insure you can resolve fmax freq.
statement of the Sampling Theorem,
from A. V. Oppenheim and A. S. Willsky,
Signals and Systems, 2nd Ed. Prentice-Hall (1996) p. 518
Nyquist what?
• Terminology: The sampling frequency of a
particular situation, which may exceed by
quite a bit the maximum frequency in the
signal, is the Nyquist frequency. Twice
the maximum frequency of the signal is
called the Nyquist rate, and is the
minimum sampling rate that can resolve
the signal. (O&W, 1st Ed, p. 519)
Resolvable frequency in spectrum
• Given the sampling rate, and the number of
samples taken, what is the Δf between points on
the spectrum?
• If a waveform is sampled for one second, what is
the frequency resolution?
• No matter how long you sample for, the maximum
reconstructable frequency remains 2*f_samp
function Test_FFT_11c(wind_flg)
Discrete Fourier Transform
• Transform vs Series?
• Computing the coefficients can take time…(bench)
• Demo with Matlab fft and ifft functions (& Lab 6 HP ‘scope)
• And the inverse Discrete Fourier Transform…
Power spectrum
• A faster algorithm, the Fast Fourier Transform
(FFT), is normally computed, for example, in
LabVIEW or MATLAB or in the math feature of the
HP 54622A digital scope.
• Power spectrum: The HH(jw) are usually complex
numbers, and their magnitudes must be taken to
find the Power Spectrum (in Matlab, abs(HH))
FFT solves a slightly different problem…
• If resolution is ∆f then the FFT stops at
Fsamp/2- ∆f
which makes sense, since what exactly
happens at Fsamp/2 itself?
• Convert FFT results to Fourier Series coeff:
An = real( FFT(n+1) ) * (2/Nsamp)
Bn = -imag( FFT(n+1) ) * (2/Nsamp)
Aliased frequencies
• Frequencies above the half the sampling rate become
aliased as lower frequencies.
• For frequencies just above half the sampling rate, up to
the sampling rate, the aliased frequency
falias = fnyq-|factual - fnyq|, a kind of mirror-image result.
• Matlab example (work\jdd\alias03.m)
• 12 Hz sinewave requires greater than 24 Hz sampling
rate to preserve correct frequency in reconstruction.
• Result of sampling 12 Hz at 20 Hz:
alias freq = 10- abs(12-10) = 8 Hz.
What happens when f_actual = f_sample?
…a triangle wave that is zero at each multiple of f_samp
Lab 6-type question: What is the alias of a 97KHz
sinewave sampled at 20KHz? What about 83KHz?
Another try at a formula for aliased frequency
where fsamp = 40 and fact = 108
Fsamp = 40
Fact = 108
frac = rem(Fact, Fsamp)
BB = floor(frac/( Fsamp/2) )
fobs = frac + BB * (Fsamp - 2*frac)
Strobe light demo
• For Mech Engineers, goal is to find “zero
frequency” and discover angular speed of
rotating machinery.
tedkinsman.photoshelter.com
About the strobe: General Radio 1531A,
from 1971. The circuit used to drive the 10
microseconds ON of the Xenon flash tube
is similar to a defibrillator: Charging a
capacitor up to several hundred volts.
Why no new strobe models on the
market? Fear of lawsuits about epilepsy,
or the high voltage?
Filtering before and after A→D conversion
•
•
•
•
Before A→D is the anti-aliasing LP filter
After: Digital (software) filtering available:
ENGN2530: Prof Silverman: Dig. Sig. Proc.
Tapped delay line: calculate “b” coefficients:
Windows to <weight> a sample of
points (or filter coefficients)
• A window is vector that weights point-for-point
(dot product) part of a sequence of data; outside
the window the sequence is set to zero.
• Rectangular window
• Cosine, Hann, Hamming windows
• For dealing with time-sample sequence that
doesn’t start and end at the same value
• http://en.wikipedia.org/wiki/Window_function#Cosine_window
number of
samples
signal
LP antialiasing
analog
filter
A-D
conversion
Hamming
window
DFT
sample rate
Spectrum