Spectral Line measurements (Forming the ACF digitally - (T x N) approach…
Spectral Line measurements
Spectral line recievers:
Receivers systems are sensitive to a range of frequencies
however to recover structure in that range of frequencies we need a special kind of receiver
Analogue filter bank
Split broad band signal dV into Nchan narrow band channels defined by post IF filters - then square law detect to give Nchan power outputs then integrate to increase the signal to noise ratio in each channel
main advantage:simple concept
Main challenge: channel-to-channel calibration - need filter and detectors characteristics should remain identical
Main disadvantage: inflexible - large Nchan requires large amounts of hardware and the channel width is fixed at construction time.
Digging weak signals out of noise
Autocorrelation is often used in the analysis of "noisy" data containing some suspected periodicities with low SNR - this is just what a spectral line "buried" in system noise is - some frequencies contribute a little more power than others
The ACF is a measure of how predictable a function is, delayed by tau, compared with the function at t
An even function of tau with maximum correlation at tau=0
A periodic function has an ACF which is periodic with the same period
sinusoids of the same omega give the same cosinusoidal ACF
Magnitude of a frequency component in ACF
Signal phase information is lost since Q(tau) defines its own starting point - i.e. not referred to fixed fiducial (reference) point which can be defined as psi =0
Forming the ACF digitally - (T x N) approach
Sample the signal at nyquist rate and store samples, separated by time intervals dt over a long time period T significantly > taumax
Cross multiply, point by point, all the samples within T, with a copy for a particular shift (an integer nu,ber x sampling interval dt)
Add up the results over all T and average to form one point in the ACF
Repeat N times taking the total shift out to tau max. Number of multiplication is alpha (NxT)
This is like a digital version a MIchelson spectral interferometer
A high SNR and oversampled periodic signal (not random) used for clarity
approach requires a long set of samples (over time period T) of f(t) to be kept together with the ability to cross multiply and add up samples with different delays -- lots of electronics needed
Only store samples of f(t) out to maximum time delay.
Cross multiply 1st point of f(t) with each of the N samples out to taumax - gives and estimate of the ACF outt totaumax at one go.
Still seeing how the signal (now only one sample) compares with shifted versions, the other single samples.
with a noise-like f(t) the SNR for each step will be low.
So use of a perfect sine wave to illustrate f(t) is deceptive
Lots of diagrams
After the next time step Dt samples move along by 1. Repeat the process and average the results after ech Dtau shift.
After collecting the same amount of data the ACF builds up to the same signal/noise as in Slide 2.
since working with fewer data at any given time this method is more efficient with electronics and is used in all DACS.
Digital Autocorrelation Spectrometer (DACS)
Single bit samples (0,1) fed into a digital delay line a "shift register") storing the sampled input signal f(t) out to N shifts
At a given time the state (0 or 1) of each stage in the shift register if compared to that of the 1st (zero shift) stage
If both the same (0 or 1): comparator 1. if different comparator 0. result added into counter for each shift
On the next clock pulse all samples move along by one stage and the process is repeate up to the chosen integration time.
The contents of the counters then form the sampled ACF
Compare all the samples of the delayed signal over a long time range T with all those of the undelayed signal for one particular shift and then repeat for N shifts out to tau max.
Complare one sampled point with single samples at all shifts out to taumax and repeat. The results are the same for same total no. of bit comparisons,
Ad/dis of dacs
Ouput accurate and stable compared with analogue systems whose characteristics tend to drift with changes in environments (temperature,mains voltage,components aging)
Noise falls off with (time)-1/2 for integration times T of many hours
Change bandwidth dv with only oone IF filter on input
change resolution dvchannel by changing digital clock rate
High res capability
increasing as electronics get faster and easier to integrate
spectroscopy of weak likes, requiring long integrations, is not a commecial application
DACS require large amounts of custom designed electronics (specialised integrated circuits)
DACS are special purpose chunks of electronics
example diagram and maths
Weiner Khinchine theorum diagram
FFTs are conceptually simpler than DACS but FFTs are computationally intensive
But commonality with commertial requirements means can use "of the shelf" integrated circuits (high speed ADCs + Field programmable Gate arrays FPGAs) --> FFT spectrometers using FPGAs are increasingly favoured over DACs