Abstract

Temporal heterodyne spectrum analysis requires the use of discrete photodetector arrays that have a large number of elements. Each element is generally followed by an amplifier, a bandpass filter, a demodulator, and nonlinear devices to handle the large dynamic range. When the number of elements in the array is of the order of 1000–2000, the readout hardware is difficult to implement. We consider decimating the array so that a much smaller number of elements are used. The spectrum is scanned across this array so that each element reads out a set of spatial frequencies in a time division multiplexing fashion. In some cases there is no penalty in dynamic range; in others, the penalty is more strongly related to the reduction in the number of photodetectors. Similar techniques are applied to a cross-spectrum analyzer that uses temporal heterodyning to derive angle of arrival information from wideband signals.

© 1988 Optical Society of America

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References

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  1. M. C. King, W. R. Bennett, L. B. Lambert, M. Arm, “Real-Time Electrooptical Signal Processors with Coherent Detection,” Appl. Opt., 6, 1367 (1967).
    [CrossRef] [PubMed]
  2. A. VanderLugt, “Interferometric Spectrum Analyzer,” Appl. Opt. 20, 2770 (1981).
    [CrossRef]
  3. M. D. Koontz, “Miniature Interferometric Spectrum Analyzer,” Proc. Soc. Photo-Opt. Instrum. Eng. 639, 126 (1986).
  4. A. VanderLugt, A. M. Bardos, “Spatial and Temporal Spectra of Periodic Functions for Spectrum Analysis,” Appl. Opt. 23, 4269 (1984).
    [CrossRef] [PubMed]
  5. A. M. Bardos, Harris Corp.

1986

M. D. Koontz, “Miniature Interferometric Spectrum Analyzer,” Proc. Soc. Photo-Opt. Instrum. Eng. 639, 126 (1986).

1984

1981

1967

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Figures (11)

Fig. 1
Fig. 1

Heterodyne spectrum analyzer.

Fig. 2
Fig. 2

Spatial frequency plane and photodetector geometry.

Fig. 3
Fig. 3

Practical implementation of heterodyne spectrum analyzer.

Fig. 4
Fig. 4

Reference waveform for scanning spectrum: (a) convergent wavefront produced by a Fresnel zone plate; (b) lens acting on a chirp waveform.

Fig. 5
Fig. 5

Dynamic range for heterodyne spectrum analyzer using a decimated array.

Fig. 6
Fig. 6

Optical subsystem to produce a decimated reference beam.

Fig. 7
Fig. 7

Dynamic range with improved reference function: curve A, fast scan mode; curve B, slow scan mode; curve C, fast scan mode with shot noise an order of magnitude less than the thermal noise; curve D, slow scan mode with shot noise an order of magnitude less than the thermal noise.

Fig. 8
Fig. 8

Dual antenna geometry to determine angle of arrival.

Fig. 9
Fig. 9

Dual channel Bragg cell processor: (a) top view; (b) side view.

Fig. 10
Fig. 10

Postdetection processing.

Fig. 11
Fig. 11

Dynamic range performance for cross-spectrum analyzer.

Equations (44)

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F ( α , t ) = - a ( x ) f ( t - x / v - T / 2 ) exp ( j 2 π α x ) d x ,
R ( α , t ) = - a ( x ) r ( t - x / v - T / 2 ) exp ( j 2 π α x ) d x .
I ( α , t ) = F ( α , t ) + R ( α , t ) exp ( j 2 π f d t ) 2 .
I 3 ( α , t ) = 2 F ( α , t ) R ( α , t ) cos ( 2 π f d t + ϕ ) ,
i 3 ( α k , t ) = A 0 cos ( 2 π f d t + ϕ ) - F ( α , t ) R ( α , t ) H ( α ) d α ,
r ( t ) = n = N 1 N 2 cos ( 2 π n f 0 t + ϕ n ) ,
c ( t ) = m = M 1 M 2 cos ( 2 π m f s t + ϕ m ) ,
ϕ m = d π m 2 M ,
c ( x ) = exp ( j π x 2 / λ D ) ,
ϕ m = ( 2 π / λ ) ( D / 2 ) m 2 θ 0 2 .
ϕ m = π λ D m 2 ( λ 2 L 2 ) = π D λ m 2 L 2 .
α max = 1 2 π x ( π x 2 λ D ) x = L / 2 = L λ D .
c ( x , t ) = Re { m exp [ j 2 π m f s ( t - x / v - T / 2 ) + j ϕ m ] } ,
c ( x , z ) = Re { m exp [ j ( k m x x + k m z z + ϕ m ) ] } ,
c ( x , z ) = Re m exp ( j k m α 0 x ) exp ( j k z ) × exp ( - j k m 2 α 0 2 z / 2 ) exp ( j π m 2 / M ) .
k m α 0 x - k m 2 α 0 2 z / 2 + π m 2 / M = 0 ,
2 x L - m λ z L 2 + m M = 0.
L 2 λ D = M .
C ( α , t ) = rect ( α / α 0 ) n = - δ [ α - ( n + t / T s ) α s ] ,
i k ( t ) = 2 A 0 n = - rect { [ a k - ( n + t / T s ) α s ] / α s } × F [ ( n + t / T s ) α s ] R * [ ( n + t / T s ) α s ] × cos { 2 π f d t + ϕ [ ( n + t / T s ) α c ] } .
SNR = i k 2 R L 2 e B ( i d ¯ + i k ¯ ) R L + 4 k T B ,
R L = 1 2 π c d f d ,
SNR = i k 2 2 e B ( i d ¯ + i k ¯ ) + 8 π k T B c d f d .
i k = δ ( 1 - γ ) i s + ( 1 - δ ) γ i r + 2 δ ( 1 - δ ) γ ( 1 - γ ) i s i r cos ( 2 π f d + ϕ k ) ,
SNR = δ ( 1 - δ ) γ ( i - γ ) i s i r 8 π k T B c d f d .
D R = 20 log ( i s max i s min ) .
SNR = δ ( 1 - δ ) γ ( 1 - γ ) i s i r 2 e B γ ( 1 - δ ) i r + 8 π k T B c d f d .
S a ( α , t ) = - a ( x ) s a ( t - x / v - T / 2 ) exp ( j 2 π α x ) d x ,
S a ( β ) = - rect ( y - H / 2 h ) exp ( j 2 π β y ) d y
= sinc ( β h ) exp ( j π β H ) ,
S a ( α , β , t ) = S a ( α , t ) S a ( β ) = S a ( α , t ) sinc ( β h ) exp ( j π β H ) .
S b ( α , t ) = S a ( α , t ) exp ( j 2 π α v τ ) ,
S b ( α , β , t ) = S a ( α , t ) exp ( j 2 π α v τ ) sinc ( β h ) exp ( - j π β H ) .
I ( α , β , t ) = S a ( α , t ) 2 sinc 2 ( β h ) { 1 + cos [ 2 π H β + ϕ ( α ) ] } .
I ( α , t ) = S a ( α , t ) + S b ( α , t ) exp ( j 2 π f d t ) 2 ,
I ( α , t ) = S a ( α , t ) 2 + S b ( α , t ) 2 + 2 { S a ( α , t ) S b * ( α , t ) cos [ 2 π f d t + ϕ ( α , t ) ] } .
I ( α , t ) = S a ( α , t ) 2 { 1 + cos [ 2 π f d t + ϕ ( α , t ) ] } ,
s a ( t ) = c j cos ( 2 π f j t + φ k ) ,
s b ( t ) = c j cos [ 2 π f j ( t - τ ) + φ k ] .
g 3 ( t ) = c j 2 cos ( 2 π f d t + ϕ j ) - A ( α - α j + α c / 2 ) 2 H ( α ) d α ,
g 3 ( t ) = A 0 c j 2 cos ( 2 π f d t + ϕ j ) ,
SN = 2 e B s ( i d ¯ + i k ¯ ) .
i k ¯ = i k S a ( α , t ) + S b ( α , t ) 2 S a ( α , t ) 2 + S b ( α , t ) 2 2 S a ( α , t ) 2 .
2 e ( i d ¯ + i k ¯ ) < 8 π k T c d f d .

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