Abstract

We outline here a new algorithm for evaluating Hankel (Fourier–Bessel) transforms numerically with enhanced speed, accuracy, and efficiency. A nonlinear change of variables is used to convert the one-sided Hankel transform integral into a two-sided cross-correlation integral. This correlation integral is then evaluated on a discrete sampled basis using fast Fourier transforms. The new algorithm offers advantages in speed and substantial advantages in storage requirements over conventional methods for evaluating Hankel transforms with large numbers of points.

© 1977 Optical Society of America

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References

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  1. D. G. Gardner, J. C. Gardner, G. Lausch, W. W. Meinke, “Method for the analysis of multi-component exponential decays,” J. Chem. Phys. 31, 987 (1959).
  2. J. Schlesinger, “Fit to experimental data with exponential functions using the fast fourier transform,” Nucl. Instrum. Methods 106, 503 (1973).
    [CrossRef]
  3. M. R. Smith, S. Cohn-Sfetcu, “Comments on fit to experimental data with exponential functions using the fast fourier transform,” Nucl. Instrum. Methods 114, 171 (1974).
    [CrossRef]
  4. S. Cohn-Sfetcu, M. R. Smith, S. T. Nichols, P. L. Henry, “A digital technique for analysing a class of multicomponent signals,” Proc. IEEE 63, 1460 (1975).
    [CrossRef]
  5. S. Cohn-Sfetcu, M. R. Smith, S. T. Nichols, “On the representation of signals by basis kernels with product argument,” Proc. IEEE 63, 326 (1975).
    [CrossRef]

1975 (2)

S. Cohn-Sfetcu, M. R. Smith, S. T. Nichols, P. L. Henry, “A digital technique for analysing a class of multicomponent signals,” Proc. IEEE 63, 1460 (1975).
[CrossRef]

S. Cohn-Sfetcu, M. R. Smith, S. T. Nichols, “On the representation of signals by basis kernels with product argument,” Proc. IEEE 63, 326 (1975).
[CrossRef]

1974 (1)

M. R. Smith, S. Cohn-Sfetcu, “Comments on fit to experimental data with exponential functions using the fast fourier transform,” Nucl. Instrum. Methods 114, 171 (1974).
[CrossRef]

1973 (1)

J. Schlesinger, “Fit to experimental data with exponential functions using the fast fourier transform,” Nucl. Instrum. Methods 106, 503 (1973).
[CrossRef]

1959 (1)

D. G. Gardner, J. C. Gardner, G. Lausch, W. W. Meinke, “Method for the analysis of multi-component exponential decays,” J. Chem. Phys. 31, 987 (1959).

Cohn-Sfetcu, S.

S. Cohn-Sfetcu, M. R. Smith, S. T. Nichols, P. L. Henry, “A digital technique for analysing a class of multicomponent signals,” Proc. IEEE 63, 1460 (1975).
[CrossRef]

S. Cohn-Sfetcu, M. R. Smith, S. T. Nichols, “On the representation of signals by basis kernels with product argument,” Proc. IEEE 63, 326 (1975).
[CrossRef]

M. R. Smith, S. Cohn-Sfetcu, “Comments on fit to experimental data with exponential functions using the fast fourier transform,” Nucl. Instrum. Methods 114, 171 (1974).
[CrossRef]

Gardner, D. G.

D. G. Gardner, J. C. Gardner, G. Lausch, W. W. Meinke, “Method for the analysis of multi-component exponential decays,” J. Chem. Phys. 31, 987 (1959).

Gardner, J. C.

D. G. Gardner, J. C. Gardner, G. Lausch, W. W. Meinke, “Method for the analysis of multi-component exponential decays,” J. Chem. Phys. 31, 987 (1959).

Henry, P. L.

S. Cohn-Sfetcu, M. R. Smith, S. T. Nichols, P. L. Henry, “A digital technique for analysing a class of multicomponent signals,” Proc. IEEE 63, 1460 (1975).
[CrossRef]

Lausch, G.

D. G. Gardner, J. C. Gardner, G. Lausch, W. W. Meinke, “Method for the analysis of multi-component exponential decays,” J. Chem. Phys. 31, 987 (1959).

Meinke, W. W.

D. G. Gardner, J. C. Gardner, G. Lausch, W. W. Meinke, “Method for the analysis of multi-component exponential decays,” J. Chem. Phys. 31, 987 (1959).

Nichols, S. T.

S. Cohn-Sfetcu, M. R. Smith, S. T. Nichols, P. L. Henry, “A digital technique for analysing a class of multicomponent signals,” Proc. IEEE 63, 1460 (1975).
[CrossRef]

S. Cohn-Sfetcu, M. R. Smith, S. T. Nichols, “On the representation of signals by basis kernels with product argument,” Proc. IEEE 63, 326 (1975).
[CrossRef]

Schlesinger, J.

J. Schlesinger, “Fit to experimental data with exponential functions using the fast fourier transform,” Nucl. Instrum. Methods 106, 503 (1973).
[CrossRef]

Smith, M. R.

S. Cohn-Sfetcu, M. R. Smith, S. T. Nichols, P. L. Henry, “A digital technique for analysing a class of multicomponent signals,” Proc. IEEE 63, 1460 (1975).
[CrossRef]

S. Cohn-Sfetcu, M. R. Smith, S. T. Nichols, “On the representation of signals by basis kernels with product argument,” Proc. IEEE 63, 326 (1975).
[CrossRef]

M. R. Smith, S. Cohn-Sfetcu, “Comments on fit to experimental data with exponential functions using the fast fourier transform,” Nucl. Instrum. Methods 114, 171 (1974).
[CrossRef]

J. Chem. Phys. (1)

D. G. Gardner, J. C. Gardner, G. Lausch, W. W. Meinke, “Method for the analysis of multi-component exponential decays,” J. Chem. Phys. 31, 987 (1959).

Nucl. Instrum. Methods (2)

J. Schlesinger, “Fit to experimental data with exponential functions using the fast fourier transform,” Nucl. Instrum. Methods 106, 503 (1973).
[CrossRef]

M. R. Smith, S. Cohn-Sfetcu, “Comments on fit to experimental data with exponential functions using the fast fourier transform,” Nucl. Instrum. Methods 114, 171 (1974).
[CrossRef]

Proc. IEEE (2)

S. Cohn-Sfetcu, M. R. Smith, S. T. Nichols, P. L. Henry, “A digital technique for analysing a class of multicomponent signals,” Proc. IEEE 63, 1460 (1975).
[CrossRef]

S. Cohn-Sfetcu, M. R. Smith, S. T. Nichols, “On the representation of signals by basis kernels with product argument,” Proc. IEEE 63, 326 (1975).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

(a) Exact analytical functions f ^(x), ĵ(x), and ĝ(x) corresponding to a uniform circular “top-hat” input function f(r) and its Airy-disk Hankel transform, including the top-hat input function f ^ (top curve); the l = 0 Bessel-function kernel ĵ (middle curve); and the Airy-disk Hankel transform ĝ (bottom curve). Parameter values for the change of variables are α = 0.01612, r0 = ρ0 = 0.06349, and β = b = 3.938, corresponding to N = 256, K1 = K2 = 4. The “dimples” in the ĝ function at larger arguments are spurious results of a plotting routine with inadequate point spacing. (b) Discrete Hankel transformation and then backtransformation of a sampled top-hat input sequence fn (top curve) into the transformed sequence gn (middle curve) and then back into a sequence fn (bottom curve), using the FHT algorithm with N = 256, K1 = K2 = 4. The portions of the transformed and backtransformed sequences gn and fn above n = 256 arise from aliasing effects in the transform algorithm and are discarded. The backtransformed sequence fn exhibits Gibbs phenomena because of the finite truncation of the gn sequence before it is backtransformed.

Fig. 2
Fig. 2

The input function and two successive FHT’s of a Laguerre–Gaussian input function with p = 8, l = 0 using the FHT algorithm with N = 128, K1 = K2 = 2. The first FHT lies essentially on top of the input function at low arguments and then drops slightly lower. The second FHT is initially slightly lower, then follows closely the first FHT. The curves are plotted as straight lines between the discrete sample points; the discrete point spacing is evident at larger arguments.

Fig. 3
Fig. 3

Results of Hankel-transforming and then back-transforming a top-hat input function f(r) = 1, 0 ≤ rb, and a uniform annular function f(r) = 1, b/4 ≤ r ≤ 3b/4, using the FHT algorithm with N = 512, K1 = K2 = 4 (α = 0.0091648, r0 = 0.0478665, b = 5.223). The slump in the top-hat response and, to a lesser extent, in the annular response at low r is related to the truncation at ρ0 of the narrow Airy-disk Hankel transform in the ρ domain; resealing of the r and ρ coordinates to give comparable widths in the two domains gives much better results.

Equations (6)

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g ( ρ ) = 2 π 0 r f ( r ) J l ( 2 π ρ r ) d r
g ^ ( y ) = - f ^ ( x ) j ^ ( x + y ) d x ,
r n = r 0 e α n ,             ρ m = ρ 0 e α m .
g m = n = 0 N - 1 f n j n + m .
g m = FFT [ FFT ( f m ) × FFT * ( j m ) ] .
N = K 2 β b ln ( K 1 β b ) , α e α N = K 1 / K 2 , r 0 ρ 0 = ( K 2 / K 1 2 ) α .

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