Pergamon Computers Elect. Engng Vol. 23, No. 3, pp. 129-134, 1997 0 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain PII: SOW-7906(%~2!3-8 0045-7906/97 s17.00 + 0.00 zyxwvut ALGORITHM FOR EFFICIENT INTERPOLATION OF REAL-VALUED SIGNALS USING DISCRETE HARTLEY TRANSFORM G. S. MAHARANA and P. K. MEHER Department of Electronic Science, Berhampur University, Bhanja Bihar, Berhampur-7, Orissa, India Abstract-A scheme for interpolation of discrete-time signals using the discrete Hartley transform (DHT), is deduced from the existing DFT-based scheme. Besides, a DHT-based fast interpolation algorithm is suggested. A comparative study of arithmetic and time complexities involved in the interpolation using different FFT and FHT algorithms is also presented. It is observed that the computational complexities of the proposed fast interpolation algorithm are signitkantly less compared with the corresponding FFT-based implementation, as well as the existing DHT-based interpolation. From computer simulation, it is observed that the present Hartley-based scheme is remarkably faster compared with the Fourier-based interpolation. 0 1997 Elsevier Science Ltd. All rights reserved. Key words: Interpolation, real-valued signal, discrete-time signal processing, digital signal processing, discrete Hartley transform, discrete Fourier transform, fast Hartley transform, fast Fourier transform. 1. INTRODUCTION Interpolation, in the context of digital signal processing, is a means to find new samples in between the available samples of a signal, so as to increase the sampling rate according to the requirement [l]. Interpolation can be used for reconstruction of synthetic speech from the signal sampled at a low sampling rate. Therefore, speech data may be sampled at a lower rate for saving the cost of storage and transmission. Interpolation also plays an important role in frequency multiplexing of SSB systems and the digital time-domain beamforming [2]. The theoretical basis for changing the sampling rate by lowpass filtering operation is proposed by Schafer and Rabiner [l] and time-domain interpolation using differentiators is introduced by Sudhakar ef al. [3]. Efficient schemes using the discrete Fourier transform (DFT) are suggested in the literature for interpolation of discrete-time signals [4-6]. Since, in most practical situations, signals are real-valued, Bracewell [7] has suggested the discrete Hartley transform (DHT) as a real-valued alternative to the DFT. Because the forward DHT is identical to the inverse DHT, only one routine be coded and stored for both forward, as well as, inverse transforms. It is reported that [8,9] the fast Hartley transform (FHT) can be used for performing convolution faster than the fast Fourier transform (FFT)-based methods [lo, 111. Further, it is shown that DHT can be used more efficiently than the DFT for extrapolation of real-valued signals [12]. In 1987, Johnson [13] has proposed a DHT-based approach for interpolation of real-valued signals. To achieve more efficient interpolation compared with the Johnson’s algorithm, as well as, the FFT-based methods, in this paper, we have developed a fast interpolation algorithm using the DHT subsequences. It is found that the proposed Hartley-based interpolation, using Radix-2 FHT [14] and Split-radix FHT (SRFHT) [14] involves significantly less arithmetic and time complexities compared with the corresponding FFT-based methods. Also it is found that, the proposed algorithm offers remarkable saving of computation over the existing Hartley-based algorithm [13]. The DHT-based scheme for interpolation is described in Section 2. The proposed fast inter- polation algorithm is deduced in Section 3. Computational complexities involved in interpolation using different FFT and FHT algorithms are discussed in Section 4. Conclusions are drawn in Section 5. 2. DEVELOPMENT OF THE DHT-BASED SCHEME FOR INTERPOLATION For developing the DHT-based scheme for interpolation, we outline some useful properties of the DHT as follows. 129