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Introduction:

 

The discrete Fourier transform (DFT) computes uniformly spaced samples in the frequency domain. The fast Fourier transform (FFT) is a computationally efficient implementation of the DFT. The chirp Z transform (CZT) allows us to efficiently compute frequency samples that are available. The CZT is then repetitively used to obtain a higher resolution by zooming onto the desired part of the spectrum. This method is widely used in the industry especially in localized high resolution spectrum analysis. It is to be noted that some technical manuals refer to this process simply as the CZT or the zoom FFT [1,2]. In practice, it is often required to carry out a series of CZTís to achieve progressively higher zooming. We will use the term ZCZT throughout this paper to distinguish this process from a single CZT. Applications of the ZCZT include ultrasonic blood flow analysis, RF Communications, mechanical stress analysis, Doppler radar, side band analysis, and modulation analysis. The importance of unequally spaced frequency samples has been discussed in the literature from as uniformly spaced over any desired arc of the unit circle with efficiency similar to that of the FFT. In various real time applications, fast hardware FFT implementation for a particular limited-length data is early as 1971 [3]. The simplest way to calculate unequally spaced frequency samples would be to calculate each and every sample individually. Such brute force evaluation is computationally intensive, requiring †complex multiplications and additions per frequency sample. In this project, we introduce the interlaced CZT to obtain unequally spaced frequency samples. It is based on the ZCZT which computes a series of CZTís over decreasing ranges to produce a denser frequency grid. However, the ZCZT does not incorporate previously computed samples and hence involves some redundant computational cost. The interlaced CZT improves on the ZCZT by carefully staggering the successive CZTís such that the newly computed frequency samples are interlaced with the previously calculated ones. This simple modification of the ZCZT efficiently produces unequally spaced frequency samples which are dense where needed and sparse elsewhere. We show that the interlaced CZT results in a significant reduction in computational complexity as compared to the regular CZT as well as the ZCZT. In addition to that, it inherits the desirable properties and applications of the CZT. The first example of such an application is converting an existing FFT algorithm that requires a power of length 2 towards computation of efficient algorithm for any DFT length. In such cases, the desired DFT is first converted to a convolution via the CZT [4]. The convolution is then computed via a fast algorithm. Since padding with zeros is possible in this case, a power of length 2 could be achieved. The above-mentioned FFT algorithm that requires a power of length 2 could then be used to compute this convolution with comparable computational efficiency. The second application is also based on the CZT converting the DFT into a convolution. This allows charge coupled devices (CCD) that implement the discrete- time FIR filters to be used to implement the DFT. This application of CCD is of particular interest in very high-frequency systems where the signals are discrete in time but not digitized.