Signal Processing

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Signal Processing

• REU talk 14jun11
• Phil Perillat

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Talk Outline

• Signals and Noise
• Properties Bandwidth, mean value, rms
• Sine and cosine functions.
• Sampling
• Nyquist theorem
• Time and frequency domain
• FFT
• Averaging
• Example
• Observing a galaxy and reducing the data.
• Summary
• Online http//www.naic.edu/phil/talks/talks.html
  ?jun11 signal procsessing reu11

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Signals and noise

• Signals are generated by some physical process
• Transitions between energy levels of an atom.
  Emits energy at a particular frequency.
• Electrons spiraling in a magnetic field emit
  energy at a range of frequencies.
• One persons signal is the other persons noise
• Hot crab nebula with crab pulsar embedded in it.
• For pulsar observer, nebula is noise to overcome
• For someone studying continuum from the nebula,
  the pulsar is noise.
• The goal is to maximize the ratio of
• (signal strength)/(noise strength)

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Noise

• Noise is a random signal
• Knowing the value of the signal at time T tells
  you nothing about the value at time T1.
• Example from nature incoherent emission between
  energy levels
• If the emission from 1 atom does not affect the
  emission from an adjacent atom, then the sum will
  be noise like even though they may all be making
  the same transition.
• Sources of noise in experiments at AO
• Thermal noise (electrons bouncing around)
• properties depend on temperature (blackbody
  spectrum)
• The physical temperature of our receivers adds
  thermal noise to the signal coming in from the
  sky. This is why we try to cool our amplifiers.
• The microwave background radiation (from the big
  bang) is present where ever we look in the sky
  and has a temp of 2.7K.

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Properties bandwidth

• How often a signal can change in 1 second.
• The bandwidth is a measure of how much
  independent information can be stored in a
  signal.
• Example
• Suppose you want to flash the lights in a room
  and your signals had bandwidths of 1 and 10 Hz.
• The 1 Hz signal would let you do it once a second
• The 10 Hz signal path would allow you do to it up
  to 10 times in a second.
• The bandwidth of a signal we measure is often
  limited by filters we place in the signal path.
• Needed to satisfy Nyquists sampling theorem
  (..coming up).

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Properties mean,rms

• Mean value. Sum N values then divide by N.
• Zero mean e.g. E field, voltage.
• Non zero mean e.g. Intensity EE always
  positive
• Sigma or rms
• Measures the spread of the values about the mean
  value.
• Sigmasqrt( (?(xi xavg)2)/(n-1) )
• Also called rms (Root Mean Square)
• When computing the rms, you need to make sure
  that the noise is sitting on a flat mean. Later
  we will see how the bandpass and ripples can make
  it hard to compute the true rms of the noise.

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Cosines and sines

• Periodic functions with properties
• Amplitude, frequency, phase, period
• Acos(wt ph) or Asin(wt-ph)
• Cos, sin take radians where 2pi radians1 cycle.
• Instead of f(Hz) use angular frequency
  w(rad/sec)
• w(rad/sec)2pif(Hz)
• Period time between peaks 1/freq.
• Phase (ph) can also be written as a sum of
  cos,sin
• Acos(wt-ph)Acos(ph)cos(wt) Asin(ph)sin(wt)
• So BAcos(ph), C.Asin(ph) then pharctan(C/B)
• Why bother?
• fitting for A,ph at fixed w Acos(wt-ph) is a non
  linear fit where Acos(ph)cos(wt)
  Asin(ph)sin(wt) is a linear fit.

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Why cosines and sines??

• Cosine and sine functions form a complete basis
  set. That means
• We can express most functions as a sum of cosines
  and sines (using the fourier transform .. coming)
• Example approximate square wave using 20 sine
  waves.
• If we show something is true for sines and
  cosines (like the sampling theorem) we are
  assured that it will work for general functions.
• We can do our analysis using sines and cosines
  and then transform back to our original
  function.
• Cos and sines are solutions to the wave equation
  which models many physical processes.

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Sampling AtoD

• Analog to Digital converter (AtoD)
• Has a max and min allowable voltage input range.
• e.g.RI /- 2.5 volts, mock Spectrometer /- .76
  volts,
• Breaks this range up into N levels and then
  converts it to digital.
• 8 bit AtoD has 28 or 256 levels
• 12 bit AtoD has 212 or 4096 levels
• You adjust the input voltage so that the noise
  input level takes up about 4 bits. The rest of
  the range is used whenever strong rfi (or
  signals) occur.

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Sampling Nyquist

• Nyquist theorem
• You need at least two samples at the highest
  frequency of your signal to reconstruct your
  analog signal without aliasing.
• Since the bandwidth tells you the highest freq.,
  sample at twice the bandwidth (or a little
  more)
• Example
• Sample a 1 Hz signal at 2 samples/second.
• Incorrectly sample a 4 Hz signal at 2
  samples/sec.
• Then connect the sampled points. what do you get

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Real versus complex sampling

• Nyquist sampling needs 2 samples at highest
  freq.
• Period of highest freq1/bandwidth
• Real sampling uses 1 AtoD converter
• Take 2 samples spaced by .5/bandwidth seconds
• Complex sampling uses 2 AtoD converters.
• Analog signal is split and the two signals are
  delayed relative to each other by 90 degrees. A
  single complex sample of the pair samples the
  signal at two different places in time. Sample
  rate is equal to the bandwidth but you need more
  hardware (twice as many AtoD converters).

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Sampling summary

• Use analog filter to band limit the signal
• Adjust AtoD input levels to give 3-4 bits on the
  noise.
• Make sure AtoD has enough bits for any large
  signals that may occur (rfi).
• Run the sampler at twice the bandwidth if real
  sampling or at the bandwidth if complex sampling.

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Time domain to Freq domain

• The Fourier transform converts signals in the
  time domain to signals in the frequency domain.
• The FFT (Fast Fourier transform) does this on
  discretely sampled finite duration data sets.
• The FFT assumes that finite duration data sets
  repeat themselves.
• The FFT is defined as
• X(k)? x(n)e2pikn/N n0..N-1
• Where to these different parts come from..

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FFT

• Sample timedt with N samples gives Total
  timeNdt
• Smallest freq has 1 cycle over total time
• f0 1/(Ndt)
• Let x(ndt) n0..N-1 be the N time samples
• Let X(kf0) be the k0..N-1 frequency channels
• X(kf0)?n0,N-1x(ndt)(cos(2pkf0ndt)isin(2p
  kf0ndt))
• 2pkf0 is one of our angular frequencies
• ndt runs through the n time samples spaced by
  dt.
• But f0dt dt/(Ndt)1/N so the dts cancel out.

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FFT

• Canceling the dts gives
• X(kf0)?n0,N-1x(ndt)(cos(2pkn/N)isin(2pkn
  /N))
• You can use complex notation
• ei ?cos(?) isin(?)
• Giving
• X(kf0)?n0,N-1x(ndt)e2pikn/N

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Why the FFT works

• If you multiply 2 cosines with different
  frequencies and then average over complete
  cycles, they average to 0.
• Earlier we said that any function could be
  constructed from a sum of cosines of different
  frequencies.
• When each freq of the FFT multplies x(t) and then
  sums over time, only components of x(t) that
  equal this frequency will be non zero. All other
  frequency components will average to 0.

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The spectral density function

• The spectral density function (or spectrum) tells
  how much energy there in each frequency channel..
• If N channels S(1..N) then
• S(n)energy in channel n
• Total power ?S(n) over the N frequency
  channels
• Radio astronomy receivers measure the E B
  fields.
• Energy is the square of the E B fields.
• When we FFT our voltage (EB field) samples we
  have frequency channels that are still in voltage
  units.
• You need to square the output of the FFT to get
  energy.
• Since the FFT gives a complex result you take the
  FFT output times its complex conjugate.

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Averaging and sigma

• Adding a constant number A to itself N times
  gives NA.
• Adding noise signals together increases the rms
  value by sqrt(N).
• Sometimes you add numbers together of opposite
  sign.
• To average numbers you sum N times and then
  divide by N.
• Constant (?1..NA )/ N A no change
• Noise (?1,,Nxi)/N Sqrt(N)/N 1/sqrt(N)
• averaging N noise samples decreases rms by
  1/sqrt(N)

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The radiometer equation

• Suppose we have a single spectrum with
• Nchan frequency channels.
• bwTot total bandwidth,
• chnBwbwTot/Nchan bandwidth of each channel
• How many independent samples are in each chan of
  1 spectra (spc)?
• chnBwbwTot/Nchan. This can change chnBw times
  per sec.
• Each spectra lasts for Nchan 1/bwTot secs so
• IndSampleschnBwtimeSpc(bwTot/Nchan)(Nchan/bw
  Tot)1
• Each spectra has 1 independent sample per channel
• If we average a spectra for 1 second we get
• Time 1spec1/chnBw
• Spectra in 1 second 1/timespec chnBw
• Averaging a spectra for 1 second has chnBw
  independent sample in each channel.

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Radiometer equation

• The relative error in our spectrum S is then
• ?S/S1/sqrt( of independent samples)
• ?S/S1/sqrt(chnBw Integrationtime)
• ?S/S1/sqrt((totBw/Nchan)time)
• Properties
• Increasing the integration time
• decreases error by 1/sqrt(time)
• Increasing the channel width (smoothing)
• decreases the error by 1/sqrt(channelWidth)

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Observing the galaxy U5852

• One second spectral average
• 1420.4058 is the hydrogen spin flip transition
• You can see our galaxy sticking up.
• 1406 Mhz Dc spike from complex sampling
• 1381 Mhz GPS L3 satellite rfi. Looking for
  nuclear explosions.
• Bandpass shape origin
• filters in our if/lo system.
• small ripples caused by signals bouncing between
  the platform and the dish (standing waves).
• Y axis units are arbitrary counts from the mock
  spectrometer. They are linear in power.

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Position switching

• Averaging the On for 300 seconds decreases the
  noise.
• We still need to remove the band pass shape.
• need to divide by a band pass correction (bpc)
  that does not include the galaxy.
• The noise in the bpc must be small enough to not
  increase the noise of the result (integration
  times for on, off should be similar).
• On, Off position switching
• On track galaxy for 300 secs
• Offgo back and track the same part of the dish.
• sky has moved, you are no longer looking at the
  galaxy.
• Removes bandpass shape from our system (IF/LO) as
  well as a large fraction of the standing waves.
• Picture 300 sec on,off. Galaxy missing from off,
  off lower than on.

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Converting to Kelvins with the cal

• We need to convert to physical energy units to do
  science.
• Spectra measured in spectrometer counts
  (spcCnts).
• The cal diode injects a known amount of noise
  into the receiver.
• The lbw cal at 1400 Mhz 8.8 Kelvins
• KperSpcCntCalDefl(inDegK)/calDefl(inSpcCnt)
• Spc(spcCnts)KperSpcCnt spc(K)
• We only integrated the cal for 10 seconds. Why
  not 150 seconds?? Adds noise to the system..
• Need to average over frequency so we dont
  increase the noise on our 300 second integrated
  spectra

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(On-off)/off in DegK

• Note (On-Off)/Off (On/Off 1)
• Subtract (on off)
• We can see the galaxy, but still has bandpass
  shape.
• Units are spectrometer units (spcU).
• Divide by off.
• Removes band pass, but changes units to off (or
  Tsys)
• To use cal conversation, we must put the
  spectrometer units back in
• Compute mean of off spectra and multiply into
  (on-off)/off

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U5852 in Jy vs vel

• Plot using Janskys and velocity
• Jansky 10-26 watts/m2/hz measures power
  received
• Velocity km/sec measures the doppler shift from
  expansion of the universe and the rotation
  velocity of the galaxy.
• What we learn of the astronomy
• Distance, lookback time, mass, rotational
  velocity, morphology (double or single peaked).
• Galaxy is also generating continuum radiation
  (offset from 0)

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Summary

• Noise comes from our equip (thermal) and the sky.
• Cos and sines can represent arbitrary functions
• Nyquist sampling requires a sample rate of at
  least 2bw
• The FFT lets you convert from the time to the
  freq. domain.
• Averaging N samples decreases noise by 1/sqrt(N)
• Radiometer equation ?T/T1/sqrt(chanBwtime)
• Position switching removes band pass and standing
  waves.
• Cals convert from spectrometer units to degK
• The spectral density function lets us measure
  physical quantities of interest.