Abstract

Coherent optical spectrum analyzers are available for determining the frequency content of many analog signals written simultaneously on film. The space bandwidth product of each signal must not exceed one thousand for a typical 35 mm system. By sacrificing the multichannel capability of such a computer, single signals with space bandwidth products greater than one million (106) can be analyzed. Two optical systems are described. The first analyzes long signals but maintains the frequency resolution of the multichannel system (one part in one thousand). The second system yields frequency resolution of at least one part in one million. Near real-time, continuous spectrum analysis of analog signals is within the current state of the art. A breadboard system is being built.

© 1966 Optical Society of America

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References

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  1. L. J. Cutrona, E. Leith, C. Palermo, J. Porcello, Inst. Radio Engrs. IT-6, 386 (1960).

1960 (1)

L. J. Cutrona, E. Leith, C. Palermo, J. Porcello, Inst. Radio Engrs. IT-6, 386 (1960).

Cutrona, L. J.

L. J. Cutrona, E. Leith, C. Palermo, J. Porcello, Inst. Radio Engrs. IT-6, 386 (1960).

Leith, E.

L. J. Cutrona, E. Leith, C. Palermo, J. Porcello, Inst. Radio Engrs. IT-6, 386 (1960).

Palermo, C.

L. J. Cutrona, E. Leith, C. Palermo, J. Porcello, Inst. Radio Engrs. IT-6, 386 (1960).

Porcello, J.

L. J. Cutrona, E. Leith, C. Palermo, J. Porcello, Inst. Radio Engrs. IT-6, 386 (1960).

Inst. Radio Engrs. (1)

L. J. Cutrona, E. Leith, C. Palermo, J. Porcello, Inst. Radio Engrs. IT-6, 386 (1960).

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Figures (10)

Fig. 1
Fig. 1

An optical multichannel spectrum analyzer.

Fig. 2
Fig. 2

Optical schematic of multichannel spectrum analyzer.

Fig. 3
Fig. 3

Optical spectrum analysis of a set of square waves. Input signals (left). Frequency display (right).

Fig. 4
Fig. 4

Scanned power spectrum of a square wave.

Fig. 5
Fig. 5

Optical schematic of two-dimensional spectrum analyzer.

Fig. 6
Fig. 6

Raster scan input format for increased time bandwidth spectrum analysis.

Fig. 7
Fig. 7

Locus of light spot in analyzer output plane as the input sinusoidal frequency is varied.

Fig. 8
Fig. 8

Block diagram of a near real-time spectrum analyzer system.

Fig. 9
Fig. 9

Comb function pulses Eq. (A-12).

Fig. 10
Fig. 10

Output light distribution in vertical dimension for a sinusoidal input signal.

Tables (1)

Tables Icon

Table B-I Vertical Spectrum Resolution for N Scan Lines

Equations (53)

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P ( u ) = n = 1 N P N ( u ) ,
P N ( u ) = | - b / 2 b / 2 S n ( x ) e 2 π i λ f u x d x | 2
F ( u , v ) f ( x , y ) e ( 2 π i / λ f ) ( u x + v y ) d x d y .
S n = S { [ x + ( 2 n - 1 ) b / 2 ] / V s } u p ( x , b ) ,
S n ( y ) = δ { y - [ b - ( 2 n - 1 ) c / 2 ] } * u p ( y , a ) ,
f ( x , y ) = n = 1 N S ( ( x + ( 2 n - 1 ) b / 2 V s ) u p ( x , b ) × { δ { y - [ b - ( 2 n - 1 ) c 2 ] } * u p ( y , a ) } ,
θ 3 db [ 2.484 / ( 2 ) 1 2 ( N - 1 ) 2 - 1 ] 1 2 .
Δ u = λ f / b .
u max = λ f k max
u max / Δ u = k max b
T a ( x , y ) = { k / E 0 ( x , y ) ] γ / 2 } ,
E 0 ( t ) = [ E i ( t ) ] - 2 / γ ,
T a ( x , y ) = { k / [ E i ( t ) - 2 / γ ] γ / 2 = k E i ( t ) .
F ( u , v ) n = 1 n = ( b / c ) S [ x + ( 2 n - 1 ) b / 2 V s ] u p ( x , b ) × { δ { y - [ b - ( 2 n - 1 ) c 2 ] } * u p ( y , a ) } e ( 2 π i / λ f ) ( u x + v y ) d x d y .
F ( u , v ) n = 1 n = ( b / c ) - Q ( x ) e 2 π i u x / λ f d x - P ( y ) e 2 π i v y / λ f d y ,
Q ( x ) Ξ S { [ x + ( 2 n - 1 ) b / 2 ] / V s } u p ( x , b )
P ( y ) Ξ δ { y - [ b - ( 2 n - 1 ) c ] / 2 } * u p ( y , a ) .
- Q ( x ) e 2 π i u x / λ f d x = b sin ( π u b / λ f ) ( π u b / λ f ) * - S [ x + ( 2 n - 1 ) b / 2 V s ] e ( 2 π i / λ f ) u x d x .
- Q ( x ) e 2 π i u x / λ f d x = b sin ( π u b / λ f ) π u b / λ f * V s exp [ - π i ( 2 n - 1 ) b u / λ f ] T ( 2 π V s u λ f ) ,
T ( ω ) Ξ - S ( t ) e i ω t d t .
- P ( y ) e 2 π i v y / λ f d y = a sin ( π v a / λ f ) ( π v a / λ f ) exp ( π i v / λ f ) [ b - ( 2 n - 1 ) c ] .
F ( u , v ) a b V s sin ( π v a / λ f ) ( π v a / λ f ) n = 1 n = ( b / c ) [ sin ( π u b / λ f ) ( π u b / λ f ) * T ( 2 π V s u λ f ) × exp - [ π i ( 2 n - 1 ) b u / λ f ] ] exp ( π i v / λ f ) [ b - ( 2 n - 1 ) c ] .
T 1 ( ω ) = δ ( ω - ω 0 ) .
F 1 ( u , v ) a b V s sin ( π v a λ f ) ( π v a λ f ) n = 1 n = ( b / c ) sin ( π u b λ f - ω 0 b 2 V s ) ( π u b λ f - ω 0 b 2 V s ) exp - i b ω 0 2 V s ( 2 n - 1 ) + i π v λ f [ b - ( 2 n - 1 ) c ]
F 1 ( u , v ) a b V s sin ( π v a λ f ) π v a λ f sin ( π u b λ f - ω 0 b 2 V s ) e i π v b / λ f ( π u b λ f - ω 0 b 2 V s ) n = 1 n = ( b / c ) exp - i ( 2 n - 1 ) [ b ω 0 2 V s + π v c λ f ] .
I ( u , v ) = F ( u , v ) 2 I ( u , v ) = ( a b V s ) 2 [ sin ( π v a λ f ) ( π v a λ f ) ] 2 [ sin ( π u b λ f - ω 0 b 2 V s ) ( π u b λ f - ω 0 b 2 V s ) ] 2 × n = 1 N e - ( 2 n - 1 ) θ n = 1 N e + i ( 2 n - 1 ) θ ,
θ Ξ ( b ω 0 / 2 V s ) + ( π v c / λ f )
n = 1 N e - i ( 2 n - 1 ) θ n = 1 N e + i ( 2 n - 1 ) θ = sin 2 ( N - 1 ) θ sin 1 θ ,
θ = π n = ( b ω 0 / 2 V s ) + ( π v max c / λ f )
v max = λ f / c [ n - ( b ω 0 / 2 V s π ) ] ,
I ( u , v ) = 2 ( a b V s ) 2 [ sin ( π v a λ f ) π v a λ f ] 2 [ sin ( π u b λ f - ω 0 b 2 V s ) π u b λ f - ω 0 b 2 V s ] 2 sin 2 ( N - 1 ) ( b ω 0 2 V s + π v c λ f ) sin 2 ( b ω 0 2 V s + π v c λ f ) ,
u = ω 0 λ f / 2 V s π .
u 0 = λ f / b ,
v 0 = λ f / a .
V max ( n ) - V max ( n + 1 ) = λ f / c .
Δ v = ( λ f b / 2 π V s c ) Δ ω 0 .
Δ u = ( λ f / 2 π V s ) Δ ω 0 .
n = 1 N e - i ( 2 n - 1 ) θ n = 1 N e i ( 2 n - 1 ) θ ,
[ e - i θ n = 0 N - 1 e - 2 i n θ ] [ e i θ n = 0 N = 1 e + 2 i n θ ]
[ ( e - i 2 ( N - 1 ) θ - 1 ) / ( e - i 2 θ - 1 ) ] [ ( e i 2 ( N - 1 ) θ - 1 ) / ( e i 2 θ - 1 ) ]
[ sin 2 ( N - 1 ) θ / sin 2 θ ] ,
[ ( sin 2 ( N - 1 ) θ 3 dB ) / ( sin 2 θ 3 dB ) ] = [ ( N - 1 ) 2 / 2 ]
sin ( N - 1 ) θ 3 dB = [ ( N - 1 ) / ( 2 ) 1 2 ] sin θ 3 dB .
( N - 1 ) θ 3 dB - [ ( N - 1 ) 3 θ 3 3 dB ] / 6 [ ( N - 1 ) / ( 2 ) 1 2 ] [ θ 3 dB - ( θ 3 3 dB / 6 ) ]
θ 3 dB [ 2.484 / ( 2 ) 1 2 ( N - 1 ) 2 - 1 ] 1 2 .
R = ( π / 2 θ 3 dB ) ,
lim N R = π N 2 ( 2.484 ) = 1.18 N
g ( x , y ) = f ( x , y ) * [ δ ( x = - b / 2 , y = c / 2 ) + δ ( x = b / 2 , y = - c / 2 ) ]
G ( u , v ) = 2 F ( u , v ) cos [ ( π / λ f ) ( b u - c v ) ] .
I D ( u , v ) = G ( u , v ) 2 = 4 F ( u , v ) 2 cos 2 [ ( π / λ f ) ( b u - c v ) ] .
( π / λ f ) ( b u max - c v max ) = π n b u max - c v max = n λ f ,
v max = 0 , u max = n λ f / b ,
v max = ± ( λ f / 2 c ) , u max = ( n λ f / b ) - ( c v max / b ) = [ ( 2 n 1 ) / 2 ] ( λ f / b ) .

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