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PROGRAMMER’S REFERENCE MANUAL

SOFTWARE SUPPORT CATEGORY
The software described in
this document is supported
by Digital Equipment Corporation under Category I
as defined on page iii of
this document.

For additional copies of this document, order No. OFr-12-AMFTA-A-D from
Software Distribution Center, Digital Equipment Corporation, Maynard,
Mass.
01754.
Price $5.00

DIGITAL

EQUIPMENT

CORPORATION

•

MAYNARD, MASSACHUSETTS

DEC- 1 2 -AMFTA-A-D
1st Printing February, 1971
2nd Printing August, 1972

1970, 1971, 1972 by Digital Equipment Corp.

The material in this handbook, including but not
limited to instruction times and operating speeds,
for information purposes and is subject to
change without notice.
is

The following are trademarks of Digital Equipment

Corporation

Maynard

Massachusetts

DEC

PDP

FLIP CHIP

DIGITAL

FOCAL
COMPUTER LAB

OMNIBUS

UNIBUS

SOFTWARE

SUPPORT

CATEGORIES

Digital Equipment Corporation (DEC) makes available four categories of software
These
categories reflect the types of support a customer may expect from DEC for a specified software
.

product.

DEC reserves the right to change the category of a software product at any time.

The four categories are as follows:

CATEGORY
Software

Products

Supported

Charge

This classification includes current versions of monitors, programming languages, and

DEC will provide installation (when applicable), advisory,
and remedial support at no charge. These services are limited to original purchasers of DEC
computer systems who have the requisite DEC equipment and software products.
support programs provided by DEC.

At the option of DEC, a software product may be recategorized from Category to
Category II for a particular customer if the software product has been modified by the customer
I

or a third party

CATEGORY
Software

Products

Receive

that

Support

for

Fee

This category includes prior versions of Category
programs and all other programs available from DEC for which support is given. Programming assistance (additional support), as
I

available, will be provided on these DEC programs and non-DEC programs when used
junction with these DEC programs and equipment supplied by DEC

con-

CATEGORY
Pre-Release

ill

Software

DEC may elect to release certain software products to customers in order to facilitate
and/or customer famil iarization
in this event, DEC will limit the use of such

final testing

pre-release software to internal, non-competitive applications
Category III software is only
supported by DEC where this support is consistent with evaluation of the software product.
.

While DEC will be grateful for the reporting of any criticism and suggestions pertaining to a
pre-release, there exists no commitment to respond to these reports.

CATEGORY
Non - Su ppor ted

Software

This category includes all programs for which no support

given

ACKNOWLEDGMENT

The PDP-12 Fast Fourier Transform + Display program is
an adaptation
of a program written by James Rothman, of Digital Equipment
Corporation.
The algorithm is described briefly in Section 7.0 of
this
manual and in detail in DECUSCOPE, Volume 7, Number
3, available
from DECUS Library, Digital Equipment Corporation, Maynard,
Massa-

chusetts

TABLE OF CONTENTS

1.0

INTRODUCTION

2.0

MINIMUM HARDWARE REQUIREMENTS

3.0

OPERATING PROCEDURE
Loading FFTD
FFTD Displays

3.1
3.2

4.0

EXAMPLE

4.1

Input Display
Transform Displays

4.2

4.2.1

Real Display

4.2.2

Imaginary Display
Magnitude Display

4.2.3

4.2.4
4.3

Real Display

4.3.2

Imaginary Display
Magnitude Display

4.3.4
5.0

1
1

5
5
6
6

7
7

Scale Factor Display
Inverse Transform Displays

4.3.1
4.3.3

8
8
8

9
9

Scale Factor Display

DATA STORAGE

6.0

DATA SCALING

7.0

SUBROUTINES USED

8.0

ALGORITHM DESCRIPTION

9. 0

CORE CHART

10.0

PROGRAM REGION DESCRIPTION

10.1

Routines

10.2

Subroutines

10.3

Symbols

10.4

Beta Registers

12
12
17
23
24

TABLE OF CONTENTS (Cont'd)

Page
11.0

ASSEMBLY INSTRUCTIONS

12. 0

SYSTEM FLOWCHARTS

13.0

PROGRAM LISTING

Index

1.0

INTRODUCTION
The FFTD

(Fast Fourier Transform + Display) program can perform a Fast
Fourier Transform or Inverse Fast Fourier Transform on 4 to 1024 real
or complex points which have been stored on a LAP6-DIAL 1 or data LINC-

tape or disk.

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

user-modified over twelve different ranges.

2. 0

MINIMUM HARDWARE REQUIREMENT S

8K PDP —

3. °

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,

where n is the tape (0-7) 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; LINC mode, I/O PRESET, and START 20
.

3.2

FFTD Displays

The first display is:

DISPLAY

SINGLE PRECISION FFT
INPUT ON DIAL UNIT? Y/N

AaP 6-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,

DISPLAY

the second display is:

UNIT NUMBER
FILE NAME

Specify the unit number, 0 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 tx.
characters long and must begin with a
non-numeric character and not contain a ?, /,
or >
\
After typing the file name, press
line PEED.
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 RFJ38 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,

DISPLAY

the second display is:

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 111.
if a ri v a vj' fJ
with a header block is on a DIAL device,
it may be accessed by t:.is
sequence (instead of the DIAL message).
The correct block number is
the value in the DIAL index plus
one.
After the file has been located,
the calculation must be specified.
*

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
2.
If there is not enough space on the DIAL tape/disk to hold the output 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:

DISPLAY 10

REPLACE?

Y/N

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

DISPLAY 11

the next display is:

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 9 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 = a^+b^.
/
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 correspond
4

to the cursor point.

The fourth word is the scope Y coordinate of the

The fourth word is a relative value and depends upon the
Because the data is scaled to nine bits
Y scale factor and Y offset.
prior to display, the fourth word or Y coordinate will range from 0001

cursor point.

1 000 0 f
o

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 1 or decreased
As the display is
Twelve different ranges are possible.
by typing Q.

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 suddenly decrease in size as significant digits are lost.

play.

If the values
The magnitude display is shown at half scale initially.
display
at full
the
allow, the number 1 can be typed once to show

scale

Pressing RETURN will cause display 12 to reappear. As many displays
Subsequent displays will be initially
as desired may be requested.
Pressing LINE FEED
shown at the same range as the preceding display.
without entering a character will cause display 1 to reappear.

4.0

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

2000

100

1000
0000

111

1000

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

Address
200

Value
2000

211
300

IMA

77 points

377
400

1000
2000
77 points

477
500

577
600
677
700
777

1000
0000
1000
2000
1000
0000

77 points

points

1000

If the input is
displayed,
look as follows, assuming

display is not moving.

there will only be a REAL
display,
It will
the cursor is to the
extreme left and the

0001
0000
2000
0601

flrSt tW ° values in the
upper le« n =
the point on which the
cursor is restina
whf,„ the cursor
is at the
extreme left it indicates
in Hinaf
location 0000 of field
The
thlrd, valu€
is the contents of
ot that memory location,
,
in this case
o aaa
f° U
value is the position
of the cursor w^h
respect to the bottom of
screen.
the
[l
[1 = bottom,
bottom Aon
4*1 = X axis (middle),
1000 = top .j
1S

4‘ 2

4’2.1

transform Displays
Real Display

0001
2000
0000
0401
6

Moving the cursor to the highest point in the display will change the
value display to: >3001
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
7

liGcausG thG magnitude of maximum values
causes overflow, a factor of 2
is removed during computation.
Therefore,

the values displayed are
half scale; type the key "1" once to display
the magnitude at full
scale

4.2.4

Scale Factor Display

The scale factor has a value of 1.
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, 0764, 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:

[(original data) *2 sum of scale factors ]/# Qf points

results -

Reducing the equation for the first point
yields:
=

[ (

10 *

2000 q
8

)

*28

]/ 1000

-2 9

4.3.2

Imaginary Display

0001
1000
0007
0401

The values are very small and are the result of imprecision in the computations

4.3.3

Magnitude Display

0001
0000
0312
0440

As in the magnitude display of the transform, the values displayed are
half scale (displayed scale factor - 1)
Because the imaginary components are essentially zero, the magnitude, when doubled, equals the
.

real values.
4.3.4

Scale Factor Display

The scale factor has a value of

7.
For the magnitude display, scale
factor is 7-1 since display is already half scale.

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
value zero.
The input and output data are in the form of binary fractions.
For output data, the location following the last imaginary part
contains the scale factor (refer to Data Scaling, Section 6.0).
A
file of complex values are stored in the following format:

-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
signed binary fractions.
The binary point is located between bit 0
and bit 1, leaving an 11 bit signed mantissa.
Bit 0 is used as a
sign bit.
Negative numbers are formed by taking the two’s complement
of the positive binary fraction, so all inputs must
be scaled in magnitude to less than one.
The outputs are also formatted as above.
In order to preserve precision,

in a computation.

it is sometimes necessary to divide
As a result, a pseudo floating point format

has been adopted in which a variable scale factor
(or exponent) is imposed on all the Fourier coefficients.
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
by 2
Because in binary notation shifting a number
.

right one bit
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~ SCAL
If, however, the inverse transform was performed on normalized transform data,
the results
are equal to tr< original data) *2 ]/no. of points Jwhere
n equals the sum
of both scale factors.
To retrieve the time samples, shift left each
number by the value of the scale factor.
is

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
.

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

8.0

ALGORITHM DESCRIPTION

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

r n-i

-Zrrijlc/N

,N-1

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
2
required times proportional to N ^ This gives a reduction in computation time of l~log N/N or over 99 percent for N=lj324.
The algorithm
.

makes use of the fact that

W k =W (kmodN) (where w=e -2Tri/N
(

}

to reduce the number of manipulations necessary for a transformation.

9 . 0

CORE CHART

Field 0

SEGMENT 0
PAGE 0
*400
*1400

SEGMENT
SEGMENT

SEGMENT

Field

IFFT
FFT
DISPLAY
MILDRED

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

2000

Buffer
Buffer

real parts
imaginary parts

01 -VI

10.0

PROGRAM REGION DESCRIPTION

10.1

Routines

IF FT

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)

070

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
5
memory for a 16 point transformation (N=16, n=4)
write 5 as a binary number of n=4 bits, 5
^=0101 2
1
Then reverse the order of these bits to
1010 2
This means S,- is stored in position 10.
Physically, then, S of the real parts is to be
found
5
in location XRTAB+9.
,

000

MULTI

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.

i&i.Q)

INVRT

Reverse the bits of the number contained in
the

Perform

AC.

TRIGET

Fetch sine and cosine values.
Specifically, if
the AC=K on entry, the values of sin
(2~trK/N) and
cos ( 2tfK/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

single precision add with rounding.

12&0

I DORA

This subroutine generates a moving window display with a cursor riding on the curve.
For

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

Display the question:

FROM DIAL UNIT?

Y/N

If the answer is Y,

to DATTAP;

A&t-iL-i

UNTFIL

—

jump to UNTFIL; if N, jump
if neither, redisplay the question.

Jump to the subroutine ASK2 to display:

UNIT NUMBER
FILE NAME

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 Description 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.

A0G1

DATTAP

Jump to the subroutine ASK3 to display:

UNIT NUMBER
BLK NUMBER
If an illegal value is entered, jump back to
DATTAP.
If all the input is legal, fall through
to MOVINP.

4063

MOVINP

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

through to PTS.

If&foA

PTS

“

Display:

NUMBER OF PTS_
(4-1024 BY PSWERS OF
REAL OR
COMPLEX? R/C_
13

Set B1 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 l'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 R0T1.

H 76

ROti

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.

STAMU

Fall

Store the power of 2 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
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.
If the answer to the
second question is not R, jump to IFCOM.
If it
is R, 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.
REALFG and jump to CKEND.

A&Z

IFCOM

Set

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

the question.

4 2 11

CKEND

If there is not enough room between
the starting

block number and the end of tape to hold
the number of points specified, jump back to PTS.
if
14

the number of output words is 400 or greater,
another block will be needed to hold the scale
factor.
Increment FDV+7. Fall through to IFFFT.

IFFFT

Display:

FFT 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
FTFLG if I set it.
If DISFLG is set, jump to
a

;

DISPLY to display the data.
OUTQES

4Z73 OUTQES

Display the question:

Otherwise, jump to

OUTPUT ON DIAL UNIT?

If the answer is Y jump to OUTUNT;

Y /N_

if N jump to

ONDAT; otherwise redisplay the question.

® OUTUNT

Jump to the subroutine AS<2 to display:

(

UNIT NUMBER^
FILE NAME
If an illegal value is input,

redisplay the quesOtherwise 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.

432$

RDDATA

Clear 4000 words of field 1 and read in the input
data.
If REALFG is 0
the data is complex - move
the imaginary parts to start at location 2000.
If
it is non-zero, the data is real and nothing need
be done.
Jump to PROC.
(

A3 o7 PROC

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

4%J

IFT

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.
ft

Jump to the subroutine FFT to transform
the input
data.
Then jump to the subroutine SORTX to sort
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.

43 b?

STSCAL

Store the scale factor in the word following
the
last imaginary part.
Move the imaginary parts from
2000 to immediately behind the real parts.

4377

nowstr

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.

4423

NOSPAC

Jump to the subroutine ASK to display the
message:

NO SPACE
HIT RETURN TO CONT

When RETURN is hit, jump to OUTQES.

4430 SAMNAM

Jump to the subroutine ASK to display:

ALREADY EXISTS
REPLACE? Y/N_

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

jump

the ques-

tion.

REPL

Try to replace the existing file with the new
file.
If the new file is longer, jump to NOSPAC.
If the replacement is successful, jump to RDDATA.

ONDAT

Jump to the subroutine ASK 3 to display:

UNIT NUMBER__
BLK NUMBER___
If an illegal value is entered,

redisplay the

question.

If there is not enough space between
the specified block number and the end of tape to

hold the output data, jump to NOSPAC.
jump to RDDATA.
10.2
'HHGG

FDV2RW

Otherwise,

Subroutines
-

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

450J

NUMBKS

Enter with the number of words in the AC.
Convert 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.

H5Z'3

ASK2

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

UNIT NUMBER
FILE NAME

by jumping to the subroutine ASK with the address of QUES2 in the AC.
Set B1 to the address

of the answer buffer and jump to the subroutine
CONV with the largest legal unit number, 17, in
the AC.
If the value is illegal, return to CALL+1.
If legal, store it and the file name in the FDV
parameter list.
Fill the file name out to 8
characters with 77* s.
Return to CALL+2.

457L

Display:

UNIT NUMBER
BLK NUMBER

by jumping to the subroutine ASK with the
address
of QUES3 in the AC.
Set B1 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
numbers are input in octal. Jump to subroutine
CONV with the largest legal unit number,
17, in
the AC.
if the value is illegal, return to CALL+l.
Otherwise, store it in word 0 of the FDV
parameter
list.
B1 is now pointing to the block number.
Jump to CONV with the largest legal block number,
777, in the AC.
If the value is illegal^, return
to CALL+l.

if legal, store it in word
FDV parameter list.
Return to CALL+2.

46 27

of the

CONV is entered with the largest legal value
in
the AC and B1 pointing to the address (1 half
word) of the first character to be converted.
Store the l's complement of the largest legal
value
in TEMP 2 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
or octal.
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
the digit being converted to it.
Repeat the procedure until a character is found which is not
between 0 and UPLEGL.
If it is not a 34, 74, or 0
xt is an illegal character:
return to CALL+l.
A
34 or 74 indicates the end of the input
field;
,

0 indicates the end of the input.

Compare the converted value in TEMPI against the maximum legal
value
in TEMP 2.
If the value is legal return to CALL+2;

otherwise return to CALL+l.
4711

OCTL

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

ASK

Q001

DISPLY

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.
Return to the
calling routine.
-

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.

£035

WCHDIS

Jump to DPIMAG, DPMAG
was

DP REAL

or DPSCAL if the answer

or S, respectively

Otherwise redisplay

the question.

£055

DPIMAG

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 IFTFLG.
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.
If IFTFLG
is non-zero, a transform was performed.
The positive half of the curve is first followed by the
negative half and the signs are reversed.
Swap
the halves and reverse signs before jumping to
PREPAR.

£117

DP REAL

Check IFTFLG for the same reason as in DPIMAG.
The only difference is that the signs of the real
parts are not reversed.
19

Q 1^0

PREPAR

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
-1000+ (1000-# of points) /2,MINPTS to the 2's

complement of the number of points, and MVDIS
to the instruction CLR,
Jump to SHOWIT.
Qt^.7

GQ1000

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 l's complement

of 1000, MINPTS to the 2 s complement of 1000
and MVDIS to the instruction SCR 4.
Fall through
f

to SHOWIT.

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 1, the lower address is always
The higher address is set in the region DISPLY.
0.
The Y offset is always 0; therefore the baseline is
half way up the scope.
The scale factor is the

instruction SCR plus the number 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.

Qlri

RFRSH

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.

G?~ H

SMALLR

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

QZtb

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.

GZZQ

DPS CAL

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.
Convert it to ASCII decimal and display it.
£270'

DP MAG

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 ques-

tion:
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

£72^

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 sum
and store it in place of the real part.
Repeat the
process for each point.
Then jump to SHOWIT to display the magnitude.

MOVPTS

¥11 G

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.
7/ 32

MVRLMG

The subroutine MVRLMG is used to swap the first
and second halves of the real or magnitude values.
In the process they are moved from the buffer start-

ing at location 0 to the one starting at 2000.

FDV

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 2 indicating
the file is binary, word 6 is the starting

block

number, and word

is the number of blocks.

Word
ENTER and REPLACE. Word 7
is filled by LOOKUP but must be supplied
for ENTER
and REPLACE.
6

is filled by LOOKUP,

Q H05~

RWPARM

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.

705 2.

SQRT

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

NU
L
S

Symbols

Number of words in computation
Power of 2 of value of N
Index to show what array is being constructed
Gives spacing between node pairs in the Lth array
Storage for N/4

NOVER4
MAXNU

Power of

MNOVR2

Storage for N/2

Pointer to real part of X(Q)
Pointer to imaginary part of X(Q)
Pointer to real part of X(P)
Pointer to imaginary part of X(P)

PR
PI

Q
P

K
C

ADD 2

TEMPR
SINE

of largest table size (13)

Numerical index Q (=0
Numerical index P (=0

1 , . . „ ,N-1)

N-l

Number in the node being operated on
Interrupts computation of Lth array every S passes
Used by subroutine ADDR as data (addend)
Used by monitor as a temporary location
Temporary storage register for real parts
Used by monitor as a temporary location
Temporary storage for sin (S*PI*K/N)
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

SHFLAG

Imaginary part of product (W *X(P)) - temporary storage
Pseudo exponent of Fourier coefficients
If =1, add with shift; if =0, add without shift

SHFCHK

Indicates if all X

DISFLG

If 00,

the data will just be displayed

IFTFLG

If 00,

an Inveise Transform was performed

REALFG

the data does not contain imaginary parts
Used to save the double precision squares of the real
and imaginary parts during calculation of the magni-

SCAL

DPSQ

in an iteration are <.5

If 00,

tude.

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 extensive
use of the Beta registers.

11.0

ASSEMBLY INSTRUCTIONS

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

—AS MILQAN,nJ
-SB MILQAN,n^
-AS SIN256 n^)
-SB SIN256,nJ
-AS FFTC- 1J
-SB FFTC-lJ
-ZE J
-AB MILQAN,nJ
-AB SIN256
J
-AB FFTC- 1 n 3
-SB FFTD,n,L J

where n is the unit
containing the program

(FFTC-1 chains to FFTC-2)

•

(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:

ANSWER
.

'DISPLAY:
UNIT NUMBER
FILE NAME
v

UNIT NUMBER
BLK NUMBER

/LEGAL
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INDEX

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

Points (cont'd)
Imaginary, 1,

Real, 1, 3,
2,

14,

17,

Core Chart, 11
Cursor, 4, 5, 6

15,

16,

14,

19,

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

Data Display,
19,

Block Number,

Replace File, 4, 16
Restart Procedure, 1,

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

Scale Factor,

16,

19,

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,
18

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

Hardware Requirements
Header Block, 2
Input Data,

12,

14,

Inverse Transform,

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,
15

14,

13,

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