1. Convolution, GPS and the TigerSHARC XCORRS instr.
M. Smith
Electrical Engineering, University of Calgary
ucalgary.ca
12/30/2018

2. Overview Correlation and the FIR Correlation and GPS
Correlation in the time domain
Correlation in the frequency domain
Some basic characteristics of the XCORRS instruction
Lab 3 – do 2 taps of real FIR filter per cycle
XCORRS – do 128 taps of complex FIR filter per cycle
12/30/2018

3. Already discussed in class for Lab. 1
You have a signal with amplitude samples A A 0 0 –A –A 0 0 A A 0 0 –A –A 0 0 A A
You have 4 FIR filters with coefficients
Which FIR produces the maximum amplitude – and when does that maximum amplitude occur?
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4. Study 1 – Review Remember to flip the coefficient order
A A 0 0 –A –A 0 0 A A 0 0 –A –A 0 0 A A Incoming signals
Output 4A
Output 2A
Output 0
Output -2A
Output -4A
Output 2A
Output 2A
Output 0
Output 0
etc
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5. Already discussed in class for Lab
Already discussed in class for Lab. 1 (Where I forget to flip coeff in examples
You have a signal with amplitude samples A A 0 0 –A –A 0 0 A A 0 0 –A –A 0 0 A A
You have 4 FIR filters with coefficients
MATCHED FILTER
MATCHED FILTER
Now imagine that the FIR filter length is 1024 – the differences between the maximums of the outputs will increase
Now imagine that the data characteristics are so that the matches between one given filter is maximized and the matches between all the other filters is minimized ORTHO-NORMAL
Now we are talking about the basics of GPS
Not talking about “FIR” any more, we are talking about “cross-correlation” – “vector products” – “inner products”. Same mathematics, different application
12/30/2018

6. GPS Number of satellites circling in the earth – 24?
Each are transmitting a unique binary signal
The binary signals are carefully chosen and carefully synchronized clocks
You have a cheap local receiver, you know the unique binary signals from each satellite
You have the FIR code from Lab. 1 – which you are going to use for cross-correlation
Laboratory #5 – determine where you are on the earth’s surface – are you really in the 515 class, or is it just a dream and you are “tucked up safe in bed” after last night’s party?
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7. GPS Positioning Concepts
(1)
For now make 2 assumptions:
We know the distance to each satellite
We know where each satellite is
With this information from 2 satellites – you know you are on a “plane of intersection.
Require 3 satellites for a 3-D position in this “ideal” scenario
Requires 4 satellites to account for local receiver clock drift.
12/30/2018

8. GPS Signal Structure
Each satellite transmits 2 carrier frequencies referred to as L1 (1575 MHz) and L2 (1227 MHz)
Each carrier frequency is BPSK modulated with a unique PRN (pseudo random number) code
The PRN code on L1 is called CA code (coarse acquisition), The PRN code on L2 is called P code (precise)
CA code takes 1 ms for full PRN transmission at 1MHz chip (bit) rate. P code takes 1.5 s for full PRN transmission at ~10MHz chip rate
Also modulated on each carrier is 50 Hz data that includes the current position of the satellite
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9. Determining Time Use the PRN code to determine time
Signal send by satellite Signal received by you
You know the signal sent
Perform correlations till you get a match
(1)
Use the PRN code to determine time
Use time to determine distance to the satellite distance = speed of light * time
12/30/2018

10. Algorithms to Find PRN Phase
Time-domain Cross correlation: 1/N ∑ x1 (n) * x2(n)
Coding equivalent to FIR filter, Lab. 1
For each of the possible 24 satellites do
For each of the N possible shift positions of the data do
For N points of data and N point of filter do complex FIR filter of N points
24 * N * N complex multiplications and additions
complex add = (A + jB) + (C + jD) = (A+ B) + j(C + D) = 2 adds complex mult = (A + jB) * (C + jD) = (AC - BC) + j(AD + BC) = 2 adds and 4 multiplications 24 * 12 * N * N operations “each time N = 1024 then that’s around 300 M ops + all the other stuff you need to do
12/30/2018

11. Algorithms to Find PRN Phase
Time-domain Cross correlation: 1/N ∑ x1 (n) * x2(n)
Coding equivalent to FIR filter, but need to filter N sets of data, each shifted by one data point
Correlation of perfectly matching signals gives a maximum value
Correlation of 2 random data sequences tends to 0
PRN code from different satellites are designed to correlate to 0. D
12/30/2018

12. Algorithms to Find PRN Phase
Time-domain Cross correlation: 1/N ∑ x1 (n) * x2(n)
Correlation in time domain equates to multiplication in frequency domain
Pre- calculate Discrete Fourier Transform of Filter coefficients X2(k) = DFT[x2(n)]
Calculate Discrete Fourier Transform of data values – once for all 24 satellites X1(k) = DFT[x1(n)] requires N log(N) complex operations
Perform multiplication in Fourier domain – requires N complex operations
Perform Inverse discrete Fourier Transform 1/N F-1[X1(k)X2(k)] -- N log(N) complex operations
Total N( log(N) ) complex operations – compared to 24 * N * N
N *N = 1024 * N log N = 1024 * log(1024) = 1024 * 10
The “long approach” is “faster” D
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13. Remember from previous lecture Know the customization issues
The satellite data has a special format 1, -1, j, -j, 1 +j, 1 –j etc, and nothing else
Essentially doing 1 bit complex arithmetic
Do the “ASP” (application specific processor) thing on the TigerSHARC
XCORRs instruction
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14. Where does the CLU fit in?
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15. THEORY Mathematical definition
XCORRS
Uses registers TR D C
And something called
CUT
THEORY Mathematical definition
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16. Standard XCORRS instruction 16 “special” correlations performed in 1 cycle 32 correlations if also using Y register
Lower 46 bits ofTHR1:0
R7:3
Quad fetches will be needed
TR0, TR1, TR2 ……. TR15
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17. TR15:0 = XCORRS(R7:4, THR3:0) TR0 += D7 * C22 + D6 * C21 +… 8 taps
………..
TR15 += D7 * C7 + D6 * C6 + … 8 taps
64 taps each cycles – on both x and y
compute blocks – if set up properly
128 taps each cycle – these are “complex taps”
compared to 2 real taps / cycle after lab. 3
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18. TR15:0 = XCORRS(R7:4, THR3:0) (CUT -7)
TR0 += D7 * C22 + D6 * C21 + … 8 taps
TR1 += D7 * C21 + D6 * C20 + … 8 taps
………..
TR14 += D7 * C8 + D6 * C taps
TR15 += D7 * C taps
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19. TR15:0 = XCORRS(R7:4, THR3:0) (CUT -15)
TR0 += D7 * C22 + D6 * C21 … 8 taps
TR1 += D7 * C21 + D6 * C20 … 7 taps
………..
TR7 += D7 * C … 1 taps
TR0 += … 0 taps
TR15 += 0 … 0 taps
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20. TR15:0 = XCORRS(R7:4, THR3:0) (CUT +15)
TR0 += 0 … taps
TR1 += D0 *C taps
………..
TR7 += D6 * C14 + D5 * C13 + … 7 taps
TR0 += D7 * C14 + D6 * C13 + … 8 taps
TR15 += D7 * C7 + D6 * C7 + … 8 taps
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21. CLU Interesting possibilities for independent study
“Q9” on final – you will always find a “make up your own question and answer it”
Bring in your head
Relevant to 4th year course
Not something we have done in the class or the labs, or presented by your colleagues (or self) as those topics are covered else where
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27. 6 OPS in 1 cycle in X – ditto in Y
MULTIPLIES
ADD REAL AND IMAG
ACCUMULATE
6 OPS in 1 cycle in X – ditto in Y
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28. Special shifter operations
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30. Bit reverse – essential for FFT operations You will find in any DSP processor
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32. Program sequencer
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35. Get the fastest hardware loop Code goes on and on
Why is all this information necessary?
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36. What exactly does all this mean?
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41. Overview Correlation and the FIR Correlation and GPS
Correlation in the time domain
Correlation in the frequency domain
Some basic characteristics of the XCORRS instruction
Lab 3 – do 2 taps of real FIR filter per cycle
XCORRS – do 128 taps of complex FIR filter per cycle
Next class – looking at using the XCORRS instruction – this was 2005 Lab. 4 exercise
12/30/2018