Abstract

The discrete Hartley transform (DHT) resembles the discrete Fourier transform (DFT) but is free from two characteristics of the DFT that are sometimes computationally undesirable. The inverse DHT is identical with the direct transform, and so it is not necessary to keep track of the +i and −i versions as with the DFT. Also, the DHT has real rather than complex values and thus does not require provision for complex arithmetic or separately managed storage for real and imaginary parts. Nevertheless, the DFT is directly obtainable from the DHT by a simple additive operation. In most image-processing applications the convolution of two data sequences f1 and f2 is given by DHT of [(DHT of f1) × (DHT of f2)], which is a rather simpler algorithm than the DFT permits, especially if images are. to be manipulated in two dimensions. It permits faster computing. Since the speed of the fast Fourier transform depends on the number of multiplications, and since one complex multiplication equals four real multiplications, a fast Hartley transform also promises to speed up Fourier-transform calculations. The name discrete Hartley transform is proposed because the DHT bears the same relation to an integral transform described by Hartley [ R. V. L. Hartley, Proc. IRE 30, 144 ( 1942)] as the DFT bears to the Fourier transform.

© 1983 Optical Society of America

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References

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  1. R. V. L. Hartley, “A more symmetrical Fourier analysis applied to transmission problems,” Proc. IRE 30, 144–150 (1942).
    [Crossref]
  2. S. Goldman, Frequency Analysis, Modulation and Noise (McGraw-Hill, New York, 1965).
  3. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).
  4. Zhong-de Wang, “Harmonic analysis with a real frequency function. I. Aperiodic case,” Appl. Math. Comput. 9, 53–73 (1981); “II. Periodic and bounded cases,”  9, 153–163 (1981); “III. Data sequence,”  9, 245–255 (1981).
    [Crossref]

1981 (1)

Zhong-de Wang, “Harmonic analysis with a real frequency function. I. Aperiodic case,” Appl. Math. Comput. 9, 53–73 (1981); “II. Periodic and bounded cases,”  9, 153–163 (1981); “III. Data sequence,”  9, 245–255 (1981).
[Crossref]

1942 (1)

R. V. L. Hartley, “A more symmetrical Fourier analysis applied to transmission problems,” Proc. IRE 30, 144–150 (1942).
[Crossref]

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).

Goldman, S.

S. Goldman, Frequency Analysis, Modulation and Noise (McGraw-Hill, New York, 1965).

Hartley, R. V. L.

R. V. L. Hartley, “A more symmetrical Fourier analysis applied to transmission problems,” Proc. IRE 30, 144–150 (1942).
[Crossref]

Wang, Zhong-de

Zhong-de Wang, “Harmonic analysis with a real frequency function. I. Aperiodic case,” Appl. Math. Comput. 9, 53–73 (1981); “II. Periodic and bounded cases,”  9, 153–163 (1981); “III. Data sequence,”  9, 245–255 (1981).
[Crossref]

Appl. Math. Comput. (1)

Zhong-de Wang, “Harmonic analysis with a real frequency function. I. Aperiodic case,” Appl. Math. Comput. 9, 53–73 (1981); “II. Periodic and bounded cases,”  9, 153–163 (1981); “III. Data sequence,”  9, 245–255 (1981).
[Crossref]

Proc. IRE (1)

R. V. L. Hartley, “A more symmetrical Fourier analysis applied to transmission problems,” Proc. IRE 30, 144–150 (1942).
[Crossref]

Other (2)

S. Goldman, Frequency Analysis, Modulation and Noise (McGraw-Hill, New York, 1965).

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).

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Figures (3)

Fig. 1
Fig. 1

Left, a waveform V(t); right, the real part of the spectrum S(ω) (dashed line), the sign-reversed imaginary part of S(ω) (dotted line), and Hartley’s transform of V(t) (solid line).

Fig. 2
Fig. 2

A 16-point representation of the truncated exponential waveform used in Fig. 1 (left) and its DHT (right).

Fig. 3
Fig. 3

Samples representing a smooth binomial hump (left) and its DHT (right).

Equations (29)

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ψ ( ω ) = ( 2 π ) - 1 / 2 - V ( t ) ( cos ω t + sin ω t ) d t ,
V ( t ) = ( 2 π ) - 1 / 2 - ψ ( ω ) ( cos ω t + sin ω t ) d ω .
S ( ω ) = ( 2 π ) - 1 / 2 - V ( t ) exp ( - i ω t ) d t
V ( t ) = ( 2 π ) - 1 / 2 - S ( ω ) exp ( i ω t ) d ω .
e ( ω ) = [ ψ ( ω ) + ψ ( - ω ) ] / 2 = ( 2 π ) - 1 / 2 - V ( t ) cos ω t d t
o ( ω ) = [ ψ ( ω ) - ψ ( - ω ) ] / 2 = ( 2 π ) - 1 / 2 - V ( t ) sin ω t d t .
S ( ω ) = e ( ω ) - i o ( ω ) = ( 2 π ) - 1 / 2 × - V ( t ) ( cos ω t - i sin ω t ) d t .
ψ ( ω ) = Re [ S ( ω ) ] - Im [ S ( ω ) ] .
V ( t ) = exp ( - t ) U ( t ) ,
S ( ω ) = ( 2 π ) - 1 / 2 ( 1 - i ω ) / ( 1 + ω 2 )
ψ ( ω ) = ( 2 π ) - 1 / 2 ( 1 + ω ) / ( 1 + ω 2 ) .
H ( ν ) = N - 1 τ = 0 N - 1 f ( τ ) cas ( 2 π ν τ / N ) ,
F ( ν ) = N - 1 τ = 0 N - 1 f ( τ ) exp ( - i 2 π ν τ / N ) .
f ( τ ) = ν = 0 N - 1 H ( ν ) cas ( 2 π ν τ / N ) .
ν = 0 N - 1 cas ( 2 π ν τ / N ) cas ( 2 π ν τ / N ) = { N , τ = τ 0 , τ τ .
ν = 0 N - 1 H ( ν ) cas ( 2 π ν τ / N ) = ν = 0 N - 1 N - 1 τ = 0 N - 1 f ( τ ) cas ( 2 π ν τ / N ) cas ( 2 π ν τ / N ) = N - 1 τ = 0 N - 1 f ( τ ) ν = 0 N - 1 cas ( 2 π ν τ / N ) cas ( 2 π ν τ / N ) = N - 1 τ = 0 N - 1 f ( τ ) × { N τ = τ 0 τ τ = f ( τ ) ,
H ( ν ) = E ( ν ) + O ( ν ) ,
E ( ν ) = [ H ( ν ) + H ( N - ν ) ] / 2
O ( ν ) = [ H ( ν ) - H ( N - ν ) ] / 2.
F ( ν ) = E ( ν ) - i O ( ν ) .
f ( τ ) = { 0.5 , τ = 0 exp ( - τ / 2 ) , τ = 1 , 2 , 15 ,
τ , ν = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 , f ( τ ) = 20 15 6 1 0 0 0 0 0 0 0 0 0 1 6 15 , H ( ν ) = 4 3.56 2.49 1.32 0 0.12 0.01 0 0 0 0 0.12 0.5 1.32 2.49 3.56.
f ( τ ) f 1 ( τ ) * f 2 ( τ ) = τ = 0 N - 1 f 1 ( τ ) f 2 ( τ - τ ) ,
H ( ν ) = H 1 ( ν ) H 2 e ( ν ) + H 1 ( - ν ) H 2 o ( ν ) ,
f 1 ( τ ) * f 2 ( τ ) = DHT of [ ( DHT of f 1 ) × ( DHT of f 2 ) ] .
H ( ν 1 , ν 2 ) = M - 1 N - 1 τ 1 = 0 M - 1 τ 2 = 0 N - 1 f ( τ 1 , τ 2 ) × cas ( 2 π ν 1 τ 1 ) cas ( 2 π ν 2 τ 2 ) , f ( τ 1 , τ 2 ) = ν 1 = 0 M - 1 ν 2 = 0 N - 1 H ( ν 1 , ν 2 ) cas ( 2 π ν 1 τ 1 ) cas ( 2 π ν 2 τ 2 ) .
DHT of f ( τ + a ) = H ( ν ) cos ( 2 π a ν / N ) - H ( - ν ) sin ( 2 π a ν / N ) .
τ = 0 N - 1 [ f ( τ ) ] 2 = N ν = 0 N - 1 [ H ( ν ) ] 2
DHT of [ τ = 0 N - 1 f 1 ( τ ) f 2 ( τ + τ ) ] = N H 1 ( ν ) H 2 ( - ν ) .