A sequence x[n] of length N has a discrete Hartley transform (DHT) defined as O...

A sequence x[n] of length N has a discrete Hartley transform (DHT) defined as

Originally proposed by R.V.L. Hartley in 1942 for the continuous-time case, the Hartley transform has properties that make it useful and attractive in the discrete-time case as well (Bracewell, 1983, 1984). Specifically, from Eq. (P8.68-1), it is apparent that the DHT of a real sequence is also a real sequence. In addition, the DHT has a convolution property, and fast algorithms exist for its computation.

In complete analogy with the DFT, the DHT has an implicit periodicity that must be acknowledged in its use. That is, if we consider x[n] to be a finite-length sequence such that x[n] = 0 for n < 0 and n > N − 1, then we can form a periodic sequence

such that x[n] is simply one period of The periodic sequence can be represented by a discrete Hartley series (DHS), which in turn can be interpreted as the DHT by focusing attention on only one period of the periodic sequence.

(a) The DHS analysis equation is defined by

Show that the DHS coefficients form a sequence that is also periodic with period N; i.e.,

(b) It can also be shown that the sequences H_N [nk] are orthogonal; i.e.,

Using this property and the DHS analysis formula of Eq. (P8.68-2), show that the DHS synthesis formula is

Note that the DHT is simply one period of the DHS coefficients, and likewise, the DHT synthesis (inverse) equation is identical to the DHS synthesis Eq. (P8.68-3), except that we simply extract one period of i.e., the DHT synthesis expression is

With Eqs. (P8.68-1) and (P8.68-4) as definitions of the analysis and synthesis relations, respectively, for the DHT, we may now proceed to derive the useful properties of this representation of a finite-length discrete-time signal.

(c) Verify that H_N [a] = H_N [a + N ], and verify the following useful property of H_N [a]:

(d) Consider a circularly shifted sequence

In other words, x₁[n] is the sequence that is obtained by extracting one period from the shifted periodic sequence Using the identity verified in part (c), show that the DHS coefficients for the shifted periodic sequence are

From this, we conclude that the DHT of the finite-length circularly shifted sequence x[((n − n₀))N ] is

(e) Suppose that x₃[n] is the N-point circular convolution of two N-point sequences x₁[n] and x₂[n]; i.e.,

By applying the DHT to both sides of Eq. (P8.68-8) and using Eq. (P8.68-7), show that

for k = 0, 1, . . . , N − 1. This is the desired convolution property. Note that a linear convolution can be computed using the DHT in the same way that the DFT can be used to compute a linear convolution. While computing X_H3[k] from X_H1[k] and X_H2[k] requires the same amount of computation as computing X₃[k] from X₁[k] and X₂[k], the computation of the DHT requires only half the number of real multiplications required to compute the DFT.

(f) Suppose that we wish to compute the DHT of an N-point sequence x[n] and we have available the means to compute the N-point DFT. Describe a technique for obtaining X= [k] from X [k] for k = 0, 1, . . . , N − 1.

(g) Suppose that we wish to compute the DFT of an N-point sequence x[n] and we have available the means to compute the N-point DHT. Describe a technique for obtaining X [k] from X_H [k] for k = 0, 1, . . . , N − 1.