Convolution, GPS and the TigerSHARC XCORRS instr.

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Convolution, GPS and the TigerSHARC XCORRS instr.

• M. Smith
• Electrical Engineering, University of Calgary
• Smithmr _at_ ucalgary.ca

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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

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Already discussed in class for Lab. 1

• You have a signal with amplitude samplesA A 0 0
  A A 0 0 A A 0 0 A A 0 0 A A
• You have 4 FIR filters with coefficients
• 0 -1 0 1 0 -1 0 1
• 0 0 -1 -1 0 0 1 1
• -1 -1 0 0 1 1 0 0
• -1 -1 1 1 1 0 0 0
• Which FIR produces the maximum amplitude and
  when does that maximum amplitude occur?

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Study 1 ReviewRemember 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
• 1 1 0 0 -1 -1 0 0
  Output 4A
• 1 1 0 0 -1 -1 0 0
  Output 2A
• 1 1 0 0 -1 -1 0 0
  Output 0
• 1 1 0 0 -1 -1 0 0
  Output -2A
• 1 1 0 0 -1 -1 0 0
  Output -4A
• A A 0 0 A A 0 0 A A 0 0 A A 0 0 A A
  Incoming signals
• 1 0 -1 0 -1 0 -1 0
  Output 2A
• 1 0 -1 0 -1 0 -1 0
  Output 2A
• 1 0 -1 0 1 0 -1 0
  Output 0
• 1 0 -1 0 1 0 -1 0
  Output 0
• etc

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Already discussed in class for Lab. 1(Where I
forget to flip coeff in examples

• You have a signal with amplitude samplesA A 0 0
  A A 0 0 A A 0 0 A A 0 0 A A
• You have 4 FIR filters with coefficients
• 0 -1 0 1 0 -1 0 1
• 0 0 -1 -1 0 0 1 1 MATCHED FILTER
• -1 -1 0 0 1 1 0 0 MATCHED FILTER
• -1 -1 1 1 1 0 0 0
• 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

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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
  earths surface are you really in the 515
  class, or is it just a dream and you are tucked
  up safe in bed after last nights party?

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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.

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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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Determining Time
Signal send by satelliteSignal 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

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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 multiplications24 12
  N N operations each time N 1024 then
  thats around 300 M ops all the other stuff
  you need to do

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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.

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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) DFTx2(n)
• Calculate Discrete Fourier Transform of data
  values once for all 24 satellitesX1(k)
  DFTx1(n) --- requires N log(N) complex
  operations
• Perform multiplication in Fourier domain
  requires N complex operations
• Perform Inverse discrete Fourier Transform 1/N
  F-1X1(k)X2(k) -- N log(N) complex operations
• Total N( 24 25 log(N) ) complex operations
  compared to 24 N N
• N N 1024 1024 --- N log N 1024
  log(1024) 1024 10
• The long approach is faster

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Remember from previous lectureKnow the
customization issues

• The satellite data has a special format1, -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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Where does the CLU fit in?
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THEORYMathematicaldefinition
XCORRS Uses registers TR D C And something
called CUT
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Standard XCORRS instruction16 special
correlations performed in 1 cycle32 correlations
if also using Y register
Lower 46 bits ofTHR10
R73
Quad fetches will be needed
TR0, TR1, TR2 . TR15
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TR150 XCORRS(R74, THR30)
TR0 D7 C22 D6 C21 8 taps TR1 D7
C21 D6 C20 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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TR150 XCORRS(R74, THR30) (CUT -7)
TR0 D7 C22 D6 C21 8 taps TR1 D7
C21 D6 C20 8 taps .. .. TR14 D7
C8 D6 C7 2 taps TR15 D7 C7
1 taps
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TR150 XCORRS(R74, THR30) (CUT -15)
TR0 D7 C22 D6 C21 8 taps TR1 D7
C21 D6 C20 7 taps .. TR7 D7 C15
1 taps TR0 0
0 taps .. TR15 0 0 taps
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TR150 XCORRS(R74, THR30) (CUT 15)
TR0 0 0
taps TR1 D0 C14 1 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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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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• MULTIPLIES
• ADD REAL AND IMAG
• ACCUMULATE
• 6 OPS in 1 cycle in X ditto in Y

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Special shifter operations
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Bit reverse essential for FFT operationsYou
will find in any DSP processor
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Program sequencer
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Get the fastest hardware loopCode goes on and on
Why is all thisinformationnecessary?
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What exactly does all this mean?
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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