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DEC-12-FQEA-D

December 1971

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OCR Text

FAST FOURIER

TRANSFORM
AND

DISPLAY

O
O
o
o

DEC-12-FQEA-D
1st Printing February, 1971

(c J

1970

Digital Equipment Corporation

The material in this handbook, including but not

limited to instruction times and operating speeds,
is

for information purposes and is subject to

change without notice.

The following are trademarks of Digital Equipment

Corporation, Maynard, Massachusetts

DEC

PDP

FLIP CHIP

FOCAL

DIGITAL

COMPUTER LAB

OMNIBUS

UNIBUS

TABLE OF CONTENTS
Page
1-0

INTRODUCTION

2.0

MINIMUM HARDWARE REQUIREMENTS

3.0

OPERATING PROCEDURE

3.1
3-2

Loading FFTD
FFTD Displays

4.0

EXAMPLE

4.1

Input Display

i
1

Transform Displays
4.2.1 Real Display
4.2

4.2.2

4.2.3
4.2.4
4.3

6
6

Imaginary Display
Magnitude Display

7
7

Scale Factor Display
Inverse Transform Displays

4.3.1

Real Display

4.3.2
4.3.3

Imaginary Display
Magnitude Display

4.3.4

Scale Factor Display

8
8
8
9
9
9

5.0

DATA STORAGE

6.0

DATA SCALING

7.0

SUBROUTINES USED

8.0

ALGORITHM DESCRIPTION

9.0

CORE CHART

10.0

PROGRAM REGION DESCRIPTION
Routines

10.1
10.2

10. 3

Subroutines
Symbols

10.4

Beta Registers

TABLE OF CONTENTS (Cont'd)

11-0

ASSEMBLY INSTRUCTIONS

12.0

SYSTEM FLOWCHARTS

13.0

PROGRAM LISTING

Index

ACKNOWLEDGMENT

The PDP-12 Fast Fourier Transform + Display program is an adaptation
of a program written by James Rothman, of Digital Equipment CorporaThe algorithm is described briefly in Section 7.0 of this
manual and in detail in DEC US COPE, Volume 72 Number 3, available
tion.

from DECUS Library, Digital Equipment Corporation, Maynard, Massachusetts

1. 0

INTRODUCTION

The FFTD

(Fast Fourier Transform

Display) program can perform a Fast

Fourier Transform or Inverse Fast Fourier Transform on
or complex points which have been stored on a LAP6-DIAL

tape or disk.

to 1024 real

or data LINC-

The real and imaginary parts of the input or output data

and the magnitude of the output data may be displayed on the scope via

Transformed data may also be stored on a DIAL or data
In addition, the scale of the displayed data can be
LINCtape or disk.
a moving window.

user-modified over twelve different ranges.

2.0

MINIMUxM HARDWARE REQUIREMENTS

8K PDP-12B with EAE.

3.0

OPERATING PROCEDURE

3.1

Loading FFTD

FFTD is a "load and go" program and is called from tape or disk by the

DIAL command:
-^LO FFTD,

n )

where n is the tape (JZf-7) or disk (10-17) containing the program. A
DIAL system tape must be on unit 0.
(If a non-existent unit is addressed,
NO is displayed on the scope.
mand.

Press RETURN and issue the proper com-

)

At any time during program operation, FFTD may be restarted by pressing the console keys; LING mode, I/O PRESET, and START 20.
3.2

FFTD Displays

The first display is:

DISPLAY 1

SINGLE PRECISION FFT
INPUT ON DIAL UNIT? Y/N

LAP6-DIAL is hereafter referred to as DIAL.
1

Type y if the data file is on a tape or disk containing DIAL; type N
if the file is on a data tape or disk,
(A file copied from paper tape

via PIP must be referenced as a data tape or disk.)
The final user replies to all the scope displays are terminated by
pressing LINE FEED.
If the input is on a DIAL tape or disk, the second display is:

DISPLAY

UNIT NUMBER^.
FILE NAME

Specify the unit number,

JZf

to 7 for tape, and 10 to 17 for disk, where

the file is located and press RETURN.
Then type the file name, which
may be 1 to 8 characters long and must begin with a non-numeric character and not contain a ?, /, \
or>
After typing the file name, press
,

LINE FEED.

Note that a file addressed by name on a DIAL tape or disk
can not have a header block and must have been placed on the device
only by the FFTD program.
If a non-existent unit is requested, NO is
displayed.

To restart the program from LINCtape, press STOP, I/O PRESET,
and START 20.
The program must be reloaded from an RK8 or RFJgfS disk.

The user is told if the file is not on the specified unit:

DISPLAY

CANNOT FIND
HIT RETURN TO CONT

Press RETURN to bring back display

If the input is on a data tape or disk, the second display is:

DISPLAY

UNIT NUMBER
BLOCK NUMBER^__

The unit may be any number from 0 to 7 for tape and 10 to 17 for
disk.
The block number must be an octal number from 0 to 777.
If a data file
with a header block is on a DIAL device, it may be accessed by this
sequence (instead of the DIAL message) .
The correct block number is
the value in the DIAL index plus one.
the calculation must be specified.

After the file has been located,

DISPLAY

HOW MANY PTS^^
(4-1024 BY POWERS OF
REAL OR
COMPLEX? R/C^

Powers of

from

to 10, are acceptable, permitting

4 to 1024 points.
Type R if the data is real; type C if it is complex.
(Refer to Section 4.0 for a description of data storage format.)
If there is not
enough room between the starting block number and the end of tape to
hold the number of points specified, display 5 will reappear.

The calculation is further specified:

DISPLAY

FFT OR DISPLAY?
TRANSFORM OR
INVERSE? T/I^

F/D

If the data is just to be displayed, type D and press RETURN,
Then
type T if the data has most recently been transformed or I if it has
not been manipulated at all or has been inversely transformed.
Con-

tinue at display

The next display is

DISPLAY

OUTPUT ON DIAL UNIT?

Y/N^

Type Y if output is to a DIAL tape or disk; type N if output is to a
data tape or disk.

A reply of Y to display
DISPLAY

(DIAL tape or disk)

causes the display:

UNIT NUMBER
FILE NAME

These answers have the same restrictions as the input display, display
If there is not enough space on the DIAL tape/disk to hold the out2.
put data, the next display is:

DISPLAY

NO SPACE
HIT RETURN TO CONT

Press RETURN to bring back display

If a file already exists with the specified name,

the next display is:

DISPIiAY 10

REPLACE?

Y/N_

Type Y or N to replace or not to replace the file.
A reply of N will
to
reappear.
8
If the file is to be replaced, but the
new file is larger than the old file, display 9 will reappear.

cause display

If output is to a data tape or disk, the next display is:

DISPLAY 11

UNIT NUMBER
BLK NUMBER^~__

The answers have the same restrictions as the input display, display 4.
If there is not enough space from the starting block number to the end
of the tape to hold the output data, display

will reappear*

The program will now read in the data, perform a Fast Fourier Transform
or Inverse Fast Fourier Transform, and write the results as complex
data pairs onto the specified tape or disk.

When the transform is completed or if just displays are desired, the
following message is displayed:
DISPLAY 12

WHICH DISPLAY?
R(EAL)
I (MAGINARY)
M(AGNITUDE)
S(CALE FACTOR)
LINE FEED (RESTART)

Type R,

I, M, or S and LINE FEED to obtain the desired display.
The
scale factor is displayed as a decimal number (0-12)
(Refer to Section 6.0, Data Scaling, for an explanation of the scale factor.) (The
.

magnitude, M, for a+b is M

= vTa^TF^r.

If the display is less than 512 points, it will be stationary and
centered on the scope.
If it contains 512 or more points, the display

can be moved in either direction using A/D knob 0.

A cursor which can be moved by rotating A/D knob 1 will ride along the
curve.
Associated with the cursor are four octal words displayed in
the top left corner of the scope, one beneath the other.
The first two
words are the absolute 15-bit core address of the cursor point. The
third word is the contents of the displayed core address, i.e., the
actual 12-bit value in the data buffer of the data word that corresponds

The fourth word is the scope Y coordinate of the

to the cursor point*

cursor point.

The fourth word is a relative value and depends upon the
y scale factor and Y offset.
Because the data is scaled to nine bits

prior to display, the fourth word or Y coordinate will range from 0001
to 1000^, where 0001 corresponds to the bottom of the scope and 1000

to the top.

The curve can be expanded in the Y direction by typing a
t>y

typing Q.

Twelve different ranges are possible.

or decreased

As the display is

enlarged, no check is made against losing significant digits of large
values because the user may wish to expand small features of the dis-

Therefore, as the display is enlarged, large values may sudden-

play.

ly decrease in size as significant digits are lost.

The magnitude display is shown at half scale initially.
allow, the number

If the values

can be typed once to show the display at full

scale.

Pressing RETURN will cause display 12 to reappear. As many displays
as desired may be requested.
Subsequent displays will be initially
shown at the same range as the preceding display.
Pressing LINE FEED

without entering a character will cause display

4.0

1 to

reappear.

EXAMPLE

This section provides examples of the displays which result from a
transform performed on a square wave of 512 points and from an inverse

transform performed on the resulting coefficients.
4 . 1

Input Display

Consider a square wave"^ of 512 real points which has the following
format on tape or disk:
Address

Value

100
111

1000

The displays shown on the following pages are adaptations and are for
demonstration purposes only.
5

Address

Value
20;?0

211

377

1000
2000

477

1000
0000

511
600

1000
2000

eii
700

1000
0000

111

1000

If the input is displayed, there will only be a REAL
display,
it will
look as follows, assuming the cursor is to the extreme
left and the
display is not moving.

0001
0000
2000
0601

The first two values in the upper left hand corner
are the address of
the point on which the cursor is resting.
When the cursor is at the
extreme left, it indicates location 0000 of field 1.
The third value
is the contents of that memory location, in
this case, 2000.
The fourth
value is the position of the cursor with respect
to the bottom of the
screen.
[1 = bottom, 401 = x axis (middle), 1000 = top.]

4.2

Transform Displays

4.2.1,

Real Display

0001
2000
0000
0401
6

Moving the cursor to the highest point in the display will change the
value display to: 0001
2400
2000
0601

This is the DC component of the wave.

4.2.2

Imaginary Display

0001
2000
0000
0401

Moving the cursor to the lowest point produces the values:
0001
2374
6567
0257

Moving the cursor to the highest point displays:
0001
2404
1214
0522

4.2.3

Magnitude Display

0001
2000
0000
0401

Moving the cursor to the highest point gives the following display:
0001
2400
1000
0501

Because the magnitude of maximum values causes overflow, a factor of
is removed during computation.
Therefore, the values displayed are
"1"
half scale; type the key
once to display the magnitude at full

scale.

4.2.4

Scale Factor Display

The scale factor has a value of

To obtain the actual coefficients,

rest the cursor on the desired point and shift right the third value
of the corner display the number of bits equal to the scale factor.
In this example, the highest value of the real display is 2000.
Shifting it right by the scale factor (=1) yields 1000, the actual value of
the DC component, which in binary is 001 000 000 000.
Because the binary point is to the right of the sign bit, the actual value is +,01^.

Inverse Transform Displays

4.3

The output of the transfer was 512 complex points.
the following displays:
4.3.1

The inverse yields

Real Display

0001
0000
0764
0477

The third value, 07 64, is a deviation from 1000, the exact value. At
this time there are 2 scale factors involved.
The relationship between
the computed results and the original data is:

factors

results = [{original data) *2

Reducing the equation for the first point yields:
1000Q =
2^ =

[(2j2f0j2fg)*2^]/1000g

21^*2^2^

=2^

points

Imaginary Display

4.3*2

0001
1000
0001
04)2(1

The values are very small and are the result of imprecision in the com--

putations.

Magnitude Display

4.3.3

0001
2000
0372
0440

As in the magnitude display of the transform, the values displayed are

half scale.

Because the imaginary components are essentially zero, the
magnitude, when doubled, equals the real values.

Scale Factor Display

4.3.4

The scale factor has a value of

5.0

DATA STORAGE

The data must be stored sequentially on tape or disk in a binary file
starting at the beginning of a block.
If the data is complex, the

real parts are grouped together followed by the imaginary parts, if
any.
If there are none, the program will create imaginary parts of

The inpat and output data are in the form of binary fracFor output data, the location following the last imaginary part

value zero.
tions.

contains the scale factor (refer to Data Scaling, Section 6.0).
file of complex values are stored in the following format:

REALi
REAL
2

IMAG,

SCALE FACTOR

-only present if file is
generated by the FFTD
program.

DATA SCALING

6.0

All calculations in FFTD are done with single precision fixed point
The binary point is located between bit 0
signed binary fractions.
and bit 1, leaving an 11 bit signed mantissa*
Bit 0 is used as a

Negative numbers are formed by taking the two's complement
of the positive binary fraction, so all inputs must be scaled in magsign bit.

nitude to les3 than one.

The outputs are also formatted as above.

In order to preserve precision, it is sometimes necessary to divide

in a computation.

As a result, a pseudo floating point format

has been adopted in which a variable scale factor (or exponent)
posed' on all the Fourier coefficients.

is im--

This scale factor or pseudo

exponent is found in item SCAL after each transform has been completed.
It is also stored after the last imaginary part on tape or disk,
The
.

values stored on tape or disk are the Fourier coefficients multiplied
SCAL
by 2
Because in binary notation shifting a number right one bit
is equivalent to dividing by two, to retrieve the coefficients themselves,
.

shift each number right by the number of bits equal to the value of the
scale factor.
In the case of the inverse transform, the time samples
are the values in memory multiplied by 2""^^^^.

If, however, the in-verse transform was performed on normalized transform data, the results
are equal to ([ (original data) *2^]/no. of points )where n equals the sum
of both scale factors.
To retrieve the time samples, shift left each
number by the value of the scale factor.

7,0

SUBROUTINES USED

Manipulation of the DIAL and data LINCtapes and disk is done using the
program MILDRED (DEC-12-FZDA)
The question and answer displays are
handled by QANDA (DEC-12-FISA)
The data displays are handled by DISPLAY
.

A modification of FFTS-C (DECUS #8-144) is used to per
form the Fourier Transforms.
(DEC-12-FLSA)

8.0

ALGORITHM DESCRIPTION

The Fast Fourier Transformation enables computation of the power specSpecifically, it permits
trum of a time series in a minimum of time.
the discrete Fourier transformation

N-1
X, e

•2Trijk/N

,N-1

i= >PT

of a series on N equally spaced time samples (where N is a power of 2)
The time required is proportional to N^log^N, whereas previous methods
This gives a reduction in computarequired times proportional to N.
The algorithm
tion time of l-log^N/N or over 99 percent for N=lj2(24.

makes use of the fact that

to reduce the number of manipulations necessary for a transformation.

9.0

CORE CHART

Field 0

SEGMENT

JZf

PAGE 0
*4j2l0

*1400

SEGMENT
SEGMENT

SEGMENT

Field

IFFT
FFT
DISPLAY
MILDRED

MONITOR
QANDA
Data display code
FDV table
RWPARM table
Questions
Sine Table
1

2000

Buffer - real parts
Buffer - imaginary parts

10.0

PROGRAM REGION DESCRIPTION

10.1

Routines

IFFT

Take the Inverse Fourier Transformation of the
data in field 1.
The results are in bit inverted

order (refer to the SORTX routine)

FFT

Take the Fourier Transformation of the data in
field 1.
The results are in bit inverted order
(refer to the SORTX routine)

SORTX

Sort the data from bit inverted order to sequential order.
Bit inversion means simply the process of re-ordering the bits in a binary number.
For instance, the binary number 001 bit inverted
is just 100 (=4).
For example, to locate S^ in
memory for a 16 point transformation (N^^'IS, n=4)

write

5 as a binary number of n=4 bits, 5^^^0101^,
Then reverse the order of these bits to IJZfljZf^.
This means S^ is stored in position 10.
Physically, then, S^ of the real parts is to be found

in location XRTAB+9.

MULTIP

INVRT

Perform a rounded single precision signed multiply
using EAE.
The CAL+1 contains the address of the
multiplicand. The AC contains the multiplier.
Exit with the product in the AC.
Reverse the bits of the number contained in the
AC.

TRIGET

Fetch sine and cosine values.
Specifically, if
the AC=K on entry, the values of sin (2irK/N) and
cos (2ifK/N) are fetched from an internal trig
table.

K must be >N/2,
A register COSINE contains the cosine value and the AC contains the
sine value on exit.

ADDR

Perform a single precision add with rounding.

This subroutine generates a moving window dis-

play with a cursor riding on the curve.

For

more information refer to the DISPLAY document,
DEC~12-FLSA-D.

Display the question:
If the answer is Y,

FROM DIAL UNIT?

Y/N_

jump to UNTFIL; if N,

jump

to DATTAP; if neither, redisplay the question.

Jump to the subroutine ASK2 to display:
UNIT NUMBER
FILE NAJ^dE

If the unit number is illegal, jump to ASK2

again to redisplay the question.

If legal, jump

to LOOKUP with the address of the File Descrip-

tion Vector (hereafter referred to as FDV) para-

meter list in the AC.
display the message:

If the file cannot be found,

CANNOT FIND
HIT RETURN TO CONT

When RETURN is hit, jump back to UNTFIL.

If the

file is found^ jump to MOVINP.

Jump to the subroutine ASK3 to display:
UNIT NUMBER
BLK NUMBER^^r
If an illegal value is entered,

DATTAP.

jump back to

If all the input is legal, fall through

to MOVINP.

Jump to FDV2RW to move the input information from
the FDV to the read/write parameter list.
Fall

through to PTS.
Display:

NUMBER OF PTS
(4-1024 BY POWERS OF
REAL OR
COMPLEX? R/C
13

Set Bl to the address of the answer buffer,
MPLIER to 12 and UPLEGL to -71 (-9) because
the number of points is entered as a decimal
value.
Set the AC to the largest legal value,
2000, and jump to CONV.

If the answer is an

illegal value jump back to PTS; store the value
in N and store its I's complement in TEMPI.
Since the number of points must be an integral

power of 2, only one bit in TEMPI may be set.
Bit 11 is the exception to one bit being a power
of 2.
Check bit 11 first, then rotate the value
adding up the number of bits set.
If the total
is not 1, jump back to PTS.
Otherwise fall through
to ROTl.

Compute the power of 2 by rotating right the
value in TEMPI and stepping B2 until the bit
that is set is encountered in bit 11,
through to STAMU,

Store the power of

Fall

in NU,
If the power is
less than 2, jump back to PTS.
Otherwise load
the AC with the number of points *2 and jump to
2

NUMBKS to compute the number of blocks needed
to hold the output.
Store the value in FDV+7.
Store it also in RWPARM+3 since, for complex
data, the input and output data consist of the
same number of blocks.
second question is not

If the answer to the
R; jump

to IFCOM.
If it
the input consists of half as many words
as the output.
Load the AC with the value of N
and jump to NUMBKS to compute the number of input blocks.
Store the value in RWPARM+3.
Set
is R,

REALPG and jump to CKEND.
If the answer is C, clear REALFG and fall through
to CKEND.
Otherwise jump back to PTS to redisplay

the question.

there is not enough room between the starting
block number and the end of tape to hold the num-

.If

ber of points specified, jump back to PTS.

the number of output words is 400 or greater,

another block will be needed to hold the scale
factor

•

Display

IFFFT

Increment FDV+7.

Fall through to IFFFT.

EFT OR DISPLAY?
TRANSFORM OR
INVERSE? T/I

F/D

If the answer to the first question is D, set

DISFLG to indicate that the data will only be
displayed.
If F, clear DISFLG to indicate that

Transform or Inverse Transform will be performed.
If the answer to the second question is T, clear

If DISFLG is set,

FTFLG; if I, set it.

DISPLY to display the data.

jump to

Otherwise, jump to

OUTQES
OUTQES

Display the question:

OUTPUT ON DIAL UNIT?

Y/N_

If the answer is Y jump to OUTUNT; if N jump to

ONDAT; otherwise redisplay the question,

OUTUNT

Jump to the subroutine Aa<2 to display;
UNIT NUMBER
FILE NAME
If an illegal value is input, redisplay the ques-

Otherwise jump to ENTER with the address of
the parameter list in the AC.
If a file with the
tion.

specified name already exists, jump to SAMNAM.
If
there is not enough space to hold the output data,
jump to NOSPAC.

If it is a new file and there is

enough space "to hold it, fall through to RDDATA.

RDDATA

Clear 4000 words of field 1 and read in the input
If REALFG is 0, the data is complex - move

data.

the imaginary parts to start at location 2000.
If
it is non-zero, the data is real and nothing need

be done.

PROC

Jump to PROC.

If IFTFLG is 0, jump to FT to do a Transform.
Otherwise, fall through to do an Inverse Transform.

Jump to the subroutine IFFT to do an Inverse
Transform on the input data.
Then jump to the

subroutine SORTX to sort the coefficients into
sequential order from bit inverted order.
Jump
to STSCAL to store the scale factor which is equal
to NU-SCAL,
The data should be shifted by this
value.

Jump to the subroutine FFT to transform the input
Then jump to the subroutine SORTX to sort
data.
the coefficients into sequential order from bit
inverted order. The scale factor is the value in

SCAL and equals the number of bits by which the
data should be shifted right.
Fall through to
STSCAL.

Store the scale factor in the word following the
last imaginary part. Move the imaginary parts from

200 to immediately behind the real parts.

Jump to the subroutine FDV2RW to move the output
parameters from the FDV to the read/write parameter
list.
Write the data onto the output tape and jump
to DISPLY.

Jump to the subroutine ASK to display the message:
NO SPACE
HIT RETURN TO CONT

When RETURN is hit, jump to OUTQES
Jump to the subroutine ASK to display:

ALREADY EXISTS
REPLACE? Y/N_,

If the answer is Y, jump to REPL; if it ia N, jump
to OUTUNT.
If it is neither, redisplay the question.

Try to replace the existing file with the new

REPL

If the new file is longer,

jump to NOSPAC,

If the replacement is successful,

jump to RDDATA.

file.

ONDAT

Jump to the subroutine ASK3 to display:

UNIT NUMBER "
BLK NUMBER
If an illegal value is entered^

question.

redisplay the

If there is not enough space between

the specified block number and the end of tape to

hold the output data, jump to NOSPAC.

Otherwise,

jump to RDDATA.
10.2

Subroutines

FDV2RW

Transfer the unit number, starting block niamber^
and number of blocks from the FDV parameter list
to the READ /WRITE parameter list.

NUMBKS

Enter with the number of words in the AC,

Con-

vert this value to blocks by counting the number
of times 400 can be subtracted from it before the
value becomes negative •
Return with the number

of blocks in the AC.
ASK2

Jump to OCTL to set MPLIER to 10 and UPLEGL to
-67 (-7) because the unit number is input as an
octal number.

UNIT NUMBER
PILE NAME

Display:

by jumping to the subroutine ASK with the ad-

dress of QUES2 in the AC.

Set Bl to the address

of the answer buffer and jump to the subroutine
CONV with the largest legal unit number, 17, in
If the value is illegal, return to CALL+1,

the AC.
If legal

store it and the file name in the FDV

parameter list.
Fill the file name out to
characters with 77»s.
Return to CALL+2.

Display:

UNIT NUMBER
BLK NUMBER

by jumping to the subroutine ASK with the address
of QUES3 in the AC.
Set Bl to the address of the

answer buffer and jump to OCTL to set MPLIER to 10
and UPLEGL to -67(7) because the unit and block
nximbers are input in octal.
Jump to subroutine

CONV with the largest legal unit number^ 17, in
If the value is illegal, return to CALL+1.
the AC.
Otherwise, store it in word 0 of the FDV parameter
list.
Bl is now pointing to the block number.
Jump to CONV with the largest legal block number,
777,

in the AC,

to CALL+1,

If the value is illegal/ return

If legal, store it in word

FDV parameter list.

of the

Return to CALL+2.

CONV is entered with the largest legal value in
the AC and Bl pointing to the address - (1 half
word) of the first character to be converted.
Store the I's complement of the largest legal value
in TEMP2 and clear TEMPI.
UPLEGL contains a -71 (-9)

or -67 (-7) and MPLIER contains a 10 or 12 depending
on whether the number to be converted is in decimal

Extract a character and compare it against
an ASCII 0 and the contents of UPLEGL,
If it is a
legal value, jump to MULPLY which will multiply the
value in TEMPI by the contents of MPLIER and add
or octal.

the digit being converted to it.

Repeat the pro-

cedure until a character is found which is not between 0 and UPLEGL.
If it is not a 34, 74, or i3,
it is an illegal character:

return to CALL+l.

34 or 74 indicates the end of the input field;

A
a

0 indicates the end of the input.
Compare the converted value in TEMPI against the maximum legal value
in TEMP2.

If the value is legal return to CALL+2;

otherwise return to CALL+1.

OCTL sets MPLIER to 10 and UPLEGL to -67 (-7) so
that CONV will convert an octal number.

ASK is entered with the address of the display in
the AC.
Store it in the parameter list and jump
to QAINIT to display the message.

Refresh the

display until the answer is input.
calling routine.

Return to the

This region is entered either after the Transform

or Inverse Transform is completed or in response
to a D in answer to the display:
FFT OR DISPLAY?
F/D_.

Since the data is manipulated in preparation for each display it must be read in before

each display.

After reading in the data, display:

WHICH DISPLAY?
R(EAL)
I (MAGINARY)

M(AGNITUDE)
S(CALE FACTOR)
LINE FEED (RESTART)
If the answer buffer contained 0, just LINE FEED
was hit:
jump to IFDIAL to restart the program.

Otherwise jump to WCHDIS.
Jump to DPIMAG, DPMAG, DP REAL, or DPSCAL if the answer
was I, M, R, or S, respectively . Otherwise redisplay
the question.
If REALFG is non-'zero,

the input is real and no

Transform was performed.
Therefore, there are no
imaginary parts to display; redisplay the question.
If REALFG is zero, check IPTFLG.

If it equals zero,

either an Inverse Transform was performed or the
original data is just being displayed.
In either
case the data is in the right order.
is non-zero,

If IFTFLG

transform was performed.

The positive half of the curve is first followed by the
negative half and the signs are reversed. Swap
a

the halves and reverse signs before jumping to
PREPAR.

Check IFTFLG for the same reason as in DPIMAG.
The only difference is that the signs of the real

parts are not reversed.
19

If less than 1000 points are to be displayed,
the display will not move and the points dis-

played will be centered on the scope.
To achieve
this, LEFTX is set to the l*s complement of
of points) /2,MINPTS to the 2's
complement of the number of points, and MVDIS
to the instruction CLR.
Jump to SHOWIT.
-10jaf0+(lj2fj30=#

If 1000 or more points are to be displayed, the
display will fill the scope and will move.
To

achieve this, LEFTX is set to the I's complement
of 1000, MINPTS to the 2's complement of 1000
and MVDIS to the instruction SCR 4.
Fall through
to SHOWIT.

Jump to the subroutine IDORA to display the data.
The six parameters following the call to IDORA are
in order:

the memory field of the lower address,
the lower address, the memory field of the higher

address, the higher address, the Y offset of the
display and the scale factor of the data. Both
fields are always
0.

the lower address is always
The higher address is set in the region DISPLY.
1,

The Y offset is always 0; therefore the baseline is
half way up the scope.
The scale factor is the
instruction SCR plus the nxuhber of bits to scale
the data right before displaying it.
Since IDORA
displays only the right nine bits, if the left
three bits are significant, the data must be scaled

right three before displaying it.
Jump to RDORA to refresh repeatedly the display
until a key on the teletype is hit.
If the RETURN
is hit, jump to REDPLY which jumps to DISPLY to
redisplay the question: WHICH DISPLAY? If a 1
is entered, jump to LARGER to blow up the display.
If a Q is hit, jump to SMALLR to decrease its size.
If anything else is entered, ignore it.

If the instruction at SIZE contains a shift of 11
bits, a bigger shift would be meaningless. Jump

back to RFRSH. Otherwise, increment the value of
the shift and jump to SHOWIT.
20

LARGER

If the instruction at SIZE contains a shift of 0
bits/ jump back to RFRSH. Otherwise decrement the

value of the shift and jump to SHOWIT.

DPSCAL

If REALFG is non-zero, only real parts are present,
meaning this program did not create the file and
therefore *there is no scale factor. Return to

DISPLY to redisplay the question.
If REALFG is
0, the scale factor is stored after the last imaginary
part.

DPMAG

Convert it to ASCII decimal and display it.

If REALFG is non-zero, the input data is real and

no transform was performed; therefore the magnitude
is the same as the real points.
Redisplay the question:

WHICH DISPLAY?

Otherwise move the imaginary
parts to location 2000.
Set RELPTR and IMGPTR, which
contain the effective address of the multipliers, to
6000 since the data begins at location 0 of their
respective segments and is fractional. Fall through
to NXTMAG.

NXTMAG

Square a real part and store it. Square the imaginary
part, add the square of the real part to it, jump to
the subroutine SQRT to get the square root of the s\im
and store it in place of the real part.
Repeat the
process for each point.
Then jump to SHOWIT to display the magnitude.

MDVPTS

The subroutine MOVPTS moves values from one buffer
(address -1 in 10) in field 1 to another (address
-1 in 11).
If CMPFLG equals 1, the values are com-

plemented as they are moved. TEMPR contains the
2's complement of the number of values to move.

MVRLMG

The subroutine MVRLMG is lised to swap the first
and second halves of the real or magnitude values.
In the process they are moved from the buffer starting at location 0 to the one starting at 2000.

The File Descriptor Vector parameter list is used
by the LOOKUP, ENTER, and REPLACE sections of MILDRED.

Word 0 contains the unit number, words 1-4 contain
21

the file name^ word 5 contains a
the file is binary, word
nuinber,
6

and word

indicating

is the starting block

is the number of blocks.

is filled by LOOKUP, ENTER and REPLACE.

Word
Word

is filled by LOOKUP but must be supplied for ENTER

and REPLACE.
The Read/Write parameter list is used by the READ
and WRITE sections of MILDRED.
Bits 0-2 of word 0
contain the field, bits 9-11 contain the unit.

Word 0 contains the starting address, word 1 the
starting tape block number and word 2 the number
of blocks.
The subroutine SQRT is entered with a value in the
double precision location DPSQ.
It returns with
the square root in the AC.

10. 3

Symbols

of words in computation

Nuitiber

Power of

Index to show what array is being constructed

Gives spacing between node pairs in the Lth array

N0VER4

Storage for N/4

MAXNU
MN0VR2
QR

Power of

Pointer to real part of X(Q)

of value of N

of largest table size (13)

Storage for N/2

Pointer to imaginary part of X(Q)

Pointer to real part of X{P)

Pointer to imaginary part of X(P)

Q
P

Numerical index Q (=0,1, . . ,N-1)
Numerical index P (=0,
• • #N-"1)

Number in the node being operated on

Interrupts computation of Lth array every

ADD2

Used by subroutine ADDR as data (addend)
Used by monitor as a temporary location

TEMPR

Temporary storage register for real parts
Used by monitor as a temporary location
Temporary storage for sin (S*PI*K/N)

SINE

, •

S passes-

Used by monitor as a temporary location

COSINE

Temporary storage for cos (2*PI*K/N)
Used by monitor as a temporary location

Real part of product (W *X(P)) - temporary storage

Used by monitor as a temporary location
GI

Imaginary part of product

SCAL

temporary storage
Pseudo exponent of Fourier coefficients

SHFLAG

If =1,

SHFCHK

Indicates if all X*s in an iteration are <.5

DISFLG

7^0,

add with shift;

(W *X(P))

if =0, add without shift

the data will just be displayed

IFTFLG

If ^0, an Inverse Transform was performed

REALFG

If ^0 , the data does not contain imaginary parts

DPSQ

Used to save the double precision squares of the real
and imaginary parts during calculation of the magnitude.

CMPFLG

If =1, the subroutine MOVPTS will complement the

values as it moves them

10. 4

Beta Registers

Beta registers

1, 2, and 3 are used by the monitor in ASK2 and ASK3 as
temporary pointers and counters.
QANDA and MILDRED make more extensiv
use of the Beta registers.

11.0

ASSEMBLY INSTRUCTIONS

The FFTD program is assembled in three sections by assembling and saving each, then adding them together
The entire command sequence is:
.

->AS
•^SE
-^AS
->SB
">AS
•^SB

MILQAN,n J

where n is the unit
containing the program

MILQAN,nJ
SIN256,n^
SIN256,nJ
FFTC-lJ
FFTC-iy

(FFTC-1 chains to FFTC-2)

->ZEj
->AB

MILQAN,n^

^AB SIN256,n J
->AB
->SB

12.0

FFTC-l,n3
FFTD,n,Lj

(saves the whole program)

SYSTEM FLOWCHARTS
(Attached)

13.0

PROGRAM LISTING
(Attached)

^ START 20 ^
READ IN DIAL I/O
ROUTINES

DISPLAY:

SINGLE PRECISION FFT
INPUT ON DIAL TAPE? Y/N

DISPLAY:
UNIT NUMBER
BLK NUMBER

''DISPLAY:

UNIT NUMBER
FILE NAME

^I40VINP ^
MOVE INPUT PARAMETERS
TO R/W PARAMETER TABLE

DISPLAY:
CANNOT FIND
HIT RETURN TO CONT/

DISPLAY:
HOW MANY PTS?
(4-102 4 BY POWERS OF
REAL OR
COMPLEX? R/C

T
25

A^Ts)

COMPUTE NUMBER
OF OUTPUT BLOCKS

ADD 1 BLOCK
FOR SCALE FACTOR
N
4-

DISPLAY:
FFT OR DISPLAY?
TRANSFORM OR
INVERSE? T/I

F/D

^ DISFLG=NON-ZERO
I

IFTFLG
= NON-ZERO

IFTFLG=j2f

^ IFDISP

^MOVE OUTPUT PARAMETERS
TO R/W PARAMETER TABLE

DISPLAY:
OUTPUT ON
DIAL UNIT?

Y/N

RDDATA

CLEAR DATA
BUFFER
READ IN
DATA

SCALE FACTOR=
SCAL

WORE AFTER LAST
P0I^fT=SCALE FACTOR
)

NOWSTR^

^ NOWSTR

PACK IMAG PARTS
BEHIND REAL
JikL

MOVE OUTPUT PARAMETERS
TO R/W PARAMETER TABLE

WRITE OUT
DATA
iDISPLY

READ IN
DATA

DISPLAY:
WHICH DISPLAY?
R(EAL)
I (MAGI NARY)
M(AGNITUDE)
S(CALE FACTOR)
LINE FEED (RESTART)

CONVERT £>CALE
FACTOR TC ASCII
DECIMAL
)

UST

YES

AL PART?

DISPLl

NO
MOVE IMAG PARTS
TO 2j3i2fj2f

SQUARE A
REAL PART

SQUAR E AN
IMAG PART

ADD SQUARES

^SQUARE ROOT

^
/

STORE IN PLACE
OF REi\L PART

YES

LEFTX = "IW+djZfjajaf-* OF PTS)/2 (I'S COMP)
MVDIS = CLR
MINPTS = COSINE

INDEX

Algorithm, 11
A/D Knob 0, 4
A/D Knob 1, 4

Points (cont'd)
Imaginary, 1, 4,

15,

16,

19,

Real, 1, 3, 4, 6, 9, 14, 15, 19,
21

Block Number,

14,

17,

Core Chart, 11
Cursor, 4, 5, 6

Program Routines, 12
Program Subroutines, 17
Program Symbols, 23
QANDA, 10

Data Display,
9,

19,

Data Scaling,
Data Storage,

Replace File, 4, 16
Restart Procedure, 1,

Disk, 1, 2, 3
DISPLAY, 10, 13

Scale Factor,

16,

19,

13, 15,

17,

Examples, 5
Imaginary Displays, 7, 9
Input Display, 5
Magnitude Displays, 7,9
Real Displays, 6, 8
Scale Factors, 8, 9
Transform Display, 6

Terminators ,2,5
Transform, 1, 3, 6, 11, 15
Unit Number, 2,
18

File Format, 9
File Name, 2, 3, 13, 15

Hardware Requirements,
Header Block, 2
Input Data,

12,

14,

la.

inverse Transform, 1, 3,
LINCtape, 1, 2, 3
Loading Procedure,

Magnitude, 1, 4, 5, 19, 21
MILDRED, 10
Moving Window, 1, 4, 13, 20
Octal Words,
Output Data,

Points
Complex, 1, 3,
15

14,

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READER^S

COMMENTS

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INDEX

Points (cont'd)
Imaginary, 1,

Algorithm, 11
A/D Knob 0, 4
A/D Knob 1, 4

4, 9, 15, 16, 19,
21
Real, 1, 3, 4, 6, 9, 14, 15, 19,
21

Block Number, 2,

17, 18

14,

Program Routines, 12
Program Subroutines 17
Program Symbols, 23
,

Core Chart, 11
Cursor, 4, 5, 6

QANDA, 10

Data Display, 3,
9,

19,

Replace File, 4, 16
Restart Procedure, 1,

Data Scaling, 5, 10
Data Storage, 9
Disk, 1, 2, 3
DISPLAY, 10, 13

Scale Factor, 4,

16,

19,

13, 15,

17,

Examples, 5
Imaginary Displays, 7, 9
Input Display, 5
Magnitude Displays, 7,9
Real Displays, 6, 8
Scale Factors, 8, 9
Transform Display, 6

Terminators ,2,5
Transform, 1, 3, 6, 11, 15
Unit Number, 2,
18

File Format, 9
File Name, 2, 3, 13, 15

Hardware Requirements
Header Block, 2
Input Data,

12,

14,

Inverse Transform, 1,
LINCtape, 1, 2, 3
Loading Procedure,

Magnitude, 1, 4, 5, 19, 21
MILDRED, 10
Moving Window, 1, 4, 13, 20
Octal Words 4
Output Data, 1,
,

Points
Complex, 1, 3,
15

14, 15

14,

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