Lecture 39
The program is as follows
.mmregs
.def _c_int00
.data
; Reserve 8 locations for x(n)
;x(n)
xn0
xn1
xn2
xn3
xn4
xn5
xn6
xn7
Q15 notation
decimal value
.word 0
; 0h
. word 16384 ; 4000h
.word 23170 ; 5A82h
. word - 24576 ; E000h
.word 12345 ; 3039h
.word 30000 ; 7530h
.word 10940 ; 2ABCh
.word 12345 ; 3039h
; Reserve 16 locations for X(k)
X0R
.word 0
X0Im .word 0
X1R .word 0
X1Im .word 0
X2R
.word 0
X2Im .word 0
X3R .word 0
X3Im .word 0
X4R
.word 0
X4Im .word 0
X5R
. word 0
X5Im .word 0
X6R
.word 0
X6Im .word 0
X7R
.word 0
;real part of X(0) =0
;imaginary part of X(0) =0
0.0
0.5
0.707
-0.25
0.3767
0.9155
0.334
0.3767
X7Im .word 0
; 8 locations for W08 to W38, twiddle factors
W08R
W08Im
W18R
W18Im
W28R
W28Im
W38R
W38Im
.word 32767
.word 0
.word 23170
.word -23170
.word 0
.word -32767
.word -23170
.word -23170
;cos(0)=1
;-sin(0)=0
;cos(pi/4)= 0.707
;-sin(pi/4)= -0.707
;cos(pi/2)= 0
;-sin(pi/2)= -1
;cos(3pi/4)= -0.707
;-sin(3pi/4)= -0.707
; 8 locations for Spectrum
S0
.word 0
S1
.word 0
S2
.word 0
S3
.word 0
S4
.word 0
S5
.word 0
S6
.word 0
S7
.word 0
;temporary locations
TEMP1
.word 0
TEMP2
.word 0
;Frequency content at 0
;Frequency content at fs/8
;Frequency content at 2fs/8
;Frequency content at 3fs/8
;Frequency content at 4fs/
;Frequency content at 5fs/8
;Frequency content at 6fs/8
;Frequency content at 7fs/8
;MAIN PROGRAM
. text
_c_int00:
SSBX SXM
; set sign extension mode bit of ST1
CALL _clear
CALL _bitrev
Clear subroutine is shown in fig. 6.11. Sixteen locations meant for final results are cleared. AR2 is
used as pointer to the locations.
Bit reverse subroutine is shown in fig. 6.12. Here, AR1 is used as pointer to x(n). AR2 is used
as pointer to X(k) locations. AR0 is loaded with 8 and used in bit reverse addressing. Instead
of N/2 =4, it is loaded with N=8 because each X(k) requires two locations, one for real part
and the other for imaginary part. Thus, x(n) is stored in alternate locations, which are meant
for real part of X(k). AR3 is used to keep track of number of transfers.
Fig. 6.11: Clear subroutine
Butterfly subroutine is invoked 12 times. Part of the subroutine is shown in fig. 6.13. Real part
and imaginary of A and B input data of butterfly structure is divided by 4 which is the scaling
factor. Real part of A data which is divided by 2 is stored in temp location. It is used further in
computation of eq (3) and eq (4) of butterfly. Division is carried out by shifting the data to the
right by two places. AR5 points to real part of A input data, AR2 points to real part of B input
data and AR3 points to real part of twiddle factor while invoking the butterfly subroutine. After
all the four equations are computed, the pointers are in the same position as they were when
the subroutine is invoked. Thus, the results are stored such that in-place computation is
achieved. Fig. 6.14 through 6.17 show the butterfly subroutine for the computation of 4
equations.
MVMM AR1,AR5
STM#TEMP1,AR4
Replace AR with AR/4
LD *AR5, -2, A
STL A, *AR5+
STL A, 1,*AR4+
Store AR/2 in TEMP1
Replace AIm with AIm/4
LD *AR5, -2, A
STL A, *AR5STL A, 1, *AR4-
Store AIm/2 in TEMP2
LD *AR2, -2, A
STL A, *AR2+
Replace BR with BR/4
LD *AR2, -2, A
STL A, *AR2-
Replace BIm with BI/4
Fig. 6.13: Butterfly Subroutine
AR5 AR2
AR3
(1) AR= AR + BR x WR –
BI x WI
Round & store
LD
MPY
MAS
ADD
ADD
STH
A= BR x W R
#0, A
*A R 2 + , *A R 3 + , A
* A R 2 -, * A R 3 , A
*A R 5 , 1 5 , A
#1, 14, A
A , 1 , *A R 5 +
A =B R x W R – B I x W I
M a k e A R 3 2 b it s a n d
A= AR + BR x W R – BI x W I
Fig. 6.14: Real part of A output of Butterfly
AR5
AR2
AR3
Fig. 6.15: Imaginary part of A output of Butterfly
;(3 ) B R= A R – (B R x W R - B I x W I)
LD
Load A with AR scaled by 2
* A R 4 + ,A
From this, subtract new AR
S U B * A R 5 + ,A
S T L A , *A R 2 +
Store lower 16 bits as BR
Fig. 6.16: Real part of B output of Butterfly
;(4) BI= AI – (BI x WR +BR x WI)
LD *AR4 -, A
SUB *AR5-, A
STL A, *AR2R
E
T
n
o
p
n
o
p
Fig. 6.17: Imaginary part of B output of Butterfly
Figure 6.18 depicts the part of the main program that invokes butterfly subroutine by supplying
appropriate inputs, A and B to the subroutine. The associated butterfly structure is also shown
for quick reference. Figures 6.19 and 6.20 depict the main program for the computation of 2nd
and 3rd stage of butterfly.
STM #X0R,AR1
STM #X1R,AR2
STM #W08R,AR3
CALL _butterfly
STM #X2R,AR1
STM #X3R,AR2
STM #W08R,AR3
CALL _butterfly
xn 0
xn 4
xn 2
xn 6
STM #X4R,AR1
STM #X5R,AR2
STM #W08R,AR3
CALL _butterfly
STM #X6R,AR1
STM #X7R,AR2
STM #W08R,AR3
CALL _butterfly
xn1
X0R
0
.
2
5
1
X1R
X2R
0
.
2
5
2
X3R
1
/
4
X4R
3
xn5
xn3
xn7
X5R
1
/
4
X6R
4
Fig. 6.18: First stage of Signal Flow graph of DITFFT
X7R
x (0)
x (4)
X0
STM #X1R,AR1
X1
X1
STM #X3R,AR2
1/
4
X2
X2
1/
4
X4
1/
4
X6
1/
4
X0
1 /4
STM #W28R,AR3
x (2)
x (6)
x (1)
x (5)
x (3)
x (7)
STM
x (0)
STM
x (4)
x (2)
STM
x (6)
W
X3
2
X3
X5
STM #X4R,AR1
X7
1
/
4
X0
1
/
4
X2
#X6R,AR2
#W08R,AR3
X1
CALL _butterfly
#X5R,AR1
X3
STM
STM
CALL _butterfly
#X7R,AR2
x (1)
x (5)
x (3)
x (7)
1
/
4
X4
1
/
4
X6
X4
1 /4
X5
X7
X5
#W28R,AR3
CALL _butterfly
X6
W
2
X7
Fig. 6.19: Second stage of Signal Flow graph of DITFFT
Fig. 6.20: Third stage of Signal Flow graph of DITFFT
After the computation of X(k), spectrum is computed using the eq(6.8). The pointer AR1 is
made to point to X(k). AR2 is made to point to location meant for spectrum. AR3 is loaded
with keeps track of number of computation to be performed. The initialization of the pointer
registers before invoking the spectrum subroutine is shown in fig. 6.21. The subroutine is
shown in fig. 6.22. In the subroutine, square of real and imaginary parts are computed and they
are added. The result is converted to Q15 notation and stored.
S ( k ) = ( X ( k )) / N
2
0 ...point
(6.8)
for AR1
k = to
N −1to X(k)
= ( X ( k ) X ( k )) / N Make
*
STM #X0R,AR1
Make AR2 to point to
spectrum location
STM #S0,AR2
STM #07,AR3
CALL _spectrum
Set a counter to 7
Invoke spectrum
subroutine
Fig. 6.21: Initialization for Spectrum Computation
_spectrum:
LD #0, A
LD #0, B
SQUR *AR1+,A
SQUR *AR1+,B
ADD B,A
STH A,1,*AR2
LD *AR2,13,A
STH A,*AR2+
BANZ
_spectrum,*AR3RET
nop
nop
Clear both accumulators
Square re
& im parts. Add them
Convert product
to 16 bit Q15 notation
Divide by 8 & store
Fig. 6.22: Subroutine for Spectrum Computation
Problem 6.8: Suggest a method to determine the length of the DFT for a sequence that is not
power of 2.
Solution: Pseudo algorithm: Let M be the length of the given sequence
•
•
•
•
A=log2(M)
B=ceil (A); select next higher integer
N=2^B
Eg.:M=9, N=16
Problem 6.9: Write pseudo code to determine 8 point DFT using DITFFT algorithm invoking Butterfly
subroutine in a nested loop for each stage.
Solution:
Stage 1
First butterfly inputs:
2nd butterfly inputs:
3rd butterfly inputs:
4th butterfly inputs:
A
X0R
X2R
X4R
X6R
B
X1R
X3R
X5R
X7R
Observation: Difference in address to A input or B input =4 (each X(k) occupies two locations,
Re & Im part)
Pseudo algorithm:
•
•
•
•
•
Stage 1:
AR1=address of X0R
AR2=address of X1R
AR3= address of W0R
Repeat 4 times
Compute 4 butterfly equations
AR1=AR1+4
AR2=AR2+4
•
End repeat
Stage 2
First butterfly inputs:
A
X0R
B
X2R
2nd butterfly inputs:
3rd butterfly inputs:
X1R
X4R
X3R
X6R
4th butterfly inputs:
X5R
X7R
Observation: difference in address to A input or B input =not constant. The difference in
address to A input or B input of the alternative butterfly=8.
Pseudo algorithm:
Stage 2:
•
•
•
•
AR1=address of X0R
AR2=address of X2R
AR3= address of W0R
Repeat 2 times
Compute 4 butterfly equations
AR1=AR1+8
AR2=AR2+8
•
•
•
•
•
End repeat
AR1=address of X1R
AR2=address of X3R
AR3= address of W2R
Repeat 2 times
Compute 4 butterfly equations
AR1=AR1+8
AR2=AR2+8
•
End repeat
Stage 3
A
B
First butterfly inputs:
X0R X4R
2nd butterfly inputs:
X1R
3rd butterfly inputs:
X2R X6R
4th butterfly inputs:
X3R
X5R
X7R
Observation: difference in address to A input or B input =2
•
•
•
•
•
•
Pseudo algorithm:
Stage 3:
AR1=address of X0R
AR2=address of X4R
AR3= address of W0R
repeat 4 times
Compute 4 butterfly equations
AR1=AR1+2
AR2=AR2+2
AR3=AR3+2
End repeat