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CALCULATIONOF THEPOLESOF 90PHASE DIFFERENCE NETWORKS [after

Heaver]

1.

SelectF

and F , the

lower

and

upper ends

of thebandwidth.

2. Calculate:

k = [1

-(

Q'= L +2L

5

+

15L

9

+ . . .

ir

a

/log

e

(Q )

3. Select a phase error thatcan be allowed, and consult the graph below to

determine

the

minimum corresponding number

of

poles

n .

4.

Choose

two

networks,

A and B. If n is

even, there

willbe n/2

poles

in

each

network. If the number of poles isodd, put an extra one in network A.

5.

Compute

the

angles

for A and the

poles

p :

J (r) =(45/n)(4r- 3) for r = 1, 2, . . .(n/2) or[(n+l)/2]

$'(r)=ARCTAN((Q

2

-Q

6

)Sin < j > (r) / [1 +(Q

2

+Q

6

)Cos 4* (r)

])

p

a

(r)

=

[l/AF /F ]TAN[

a

(r)

- r)]

and the angles for B with its poles p, :

*

b

r)=(45/n)(4r- 1) for r = 1, 2, ... (n/2)or[(n-l)/2]

^ r)=

ARCTAN[(Q

2

-Q

6

)

Sin4 r)/ [1 +(Q

2

+Q

6

)Cos4

b

(r)])

P

b

(r)

=

[l//(F

/F

h

)]

TAN[((.

b

(r)

-

*

b

(r)]

6. The

above poles

are

normalized

in

terms

of F . To get the

a ctual poles

for

thenetworks,multiplyby F .

6a (14)

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Phase

Error

11

Bandwidth f./fJ

10 ' 100

1000

1 ,

0.1

0.01

11

DesignChart

for

Number

of

Poles

(n) for a

Given Normalized Bandwidth

andAllowable Phase

Error.

The Unwanted Sideband Rejection is also

Plotted along with

the

PhaseError. [After Bedrosian]

DETERMINING THEREQUIRED ACCURACY:

Whenit comes to determining the required accuracy of multipliers and90PDN

the two major considerations are carrier rejection and unwanted sideband

rejection. Carrier rejectionislarglyamatterof thepropertiesof themulti-

pliers. This comes inwhenyou apply two signals f andf to a multiplier and

find that in addition tof-

L

+f

2

and f. f

you also get a little and a littlef,.

Generally, thehigherof the two frequencies willbe the one thatisnoticed,and

this is the

reason

for the

term carrier rejection.

The

point

is

that

if

this

get

through

the

multiplier,

it

will find

its way to the

output

of the

frequency shifte

Forexample,if it is theprogram

signal

thatis thehighe st, this will come throu

the

multiplier

as a

Sin(u

t +

4.)

in one

case

and a

Cos(u

t

+< >)

in the

other case

where i f is thephase shiftacrossthe multiplie r which is the same for eitherof

themultipliers and foreitherof thequadr ature signals since theyare the same

frequency. There isnothing thatthesummers do

thatwill

get rid of theprogram

signal onceit isthrough, since theyare inquadrature,and nocombinationof

additionor subtraction will cause them tocancel. Thus, carrier rejection is

important inmultiplier selectionforfrequency Shifters.

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Normalized Pole Positions

for

bandwidthf./f.

=

1000. No te

the

symmetry

thatshowsup inthisplotofLog(Polepositions)

vs.

Linearn. The poles

were calculated

according to

Weaver'smethod.

6a (16)