# Expository Essay on Scallop Loss and Spectral Leakage

2021-05-12
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## Introduction

By the use of the DFT (Discrete Fourier Transform) through FFT, FT (of the finite length signal) can be incorporated so as to examine the content of the signal frequency. This strategy is used in the operation of the digital spectrum (Manolaskis & Proakis, 1996). The DFT is a transform that deals with the discrete number of frequencies or the finite discrete-time.

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For the signal that is time-restricted to order numbers 0, 1, 2, 3, L-1. N greater than or equal to L frequencies are assumed to pertain to all the data in the signal, that is, x (n) can be recovered from.

## What Is an Interpolation?

The interpolation is an essential aspect of Fourier analysis that allows users in increasing the resolution of vision of the spectrum. However, the interpolation does not give novel information. For example, when an interpolation factor is 1, the N points comprising of the time signal will be transformed into precisely N DFT bins by the use of the FFT, having a resolution of [fs/N]. Also, when the interpolation factor is 5, N sample of the time function will possess 4N zeroes appended, thus making FFT produce 5N DFT bins, having a resolution of [fs/5N]. However, interpolation cannot play a role in solving nearby or adjacent frequencies. The interpolation can make it easy in resolving visually the maximum of the isolated frequency that has no essential noise or signal in the spectrum. Statistically, the privileged density of the FFT result bin will most likely make the maximum magnitude bin closer to the frequency of the isolated frequency of the sinusoid (Manolaskis & Proakis, 1996).

## Window Functions and Spectral Estimation

The results of the spectral leakage can be minimized by the reduction of the discontinuities at the ends of the signal dimension period. This aspect leads to the notion of multiplying the signal within the dimension period by some function that easily minimizes the signal to zero at the ending points, thus evading discontinuities. The process of multiplying the data of the signal by a function that easily approaches zero at the endpoints is referred to as the window function. Analyzing the effect of the window function is done by convolving the spectrum of the signal and spectrum leakage can be noticed by spreading of the frequency elements (Fessler, 2004). For example, when a rectangular window is used by default, it is required to exhibit a peak Scallop Loss of 3.98 dBs. The estimation is about 13.6 dBs between the maximum point of the main lobe and the maximum point of the first side lobe. All alternative windows will minimize the Scallop and minimize the impact of the spectral leakage. Besides, the signal comprising of various sinusoidal elements of different amplitudes with all phases being zero will have individual frequencies estimated to an accuracy of 1Hz with an estimation of the amplitude.

## FFT Applications and Time-Frequency Signal Processing

### FFT for Convolution and Interpolation: What Is Fast Fourier Transform?

The convolution of two sets of data is a process that can be adopted in various kinds of signal processing, edge detection, or data smoothing. The main objective of the convolution process is to involve the impact of the signal in the system. For example, when we use FFT, it is essential to understand that the convolution process becomes a cyclic convolution course of action. This means that the response signal to be processed must be considered as the periodic signal. Also, in the spectrum modeling of any audio signal, we deal with indefinite period signals. Fourier analysis and indefinite period signals are performed by the adoption of the Discrete-Time Fourier Transform (DTFT), and it is feasible to interpolate a time signal by rightfully appending zeros into the spectrum of the periodic signal (Ondrej & Martin, 2006).

### Time-Frequency: What Is Non-Stationary Signal?

A non-stationary signal brushes away the frequency signal from low to high frequencies or from high to low frequencies linearly and quadratic. The first and essential way to produce a chirp signal is to concatenate a sequence of segments of the sine waves, with each sine wave escalating or decreasing in frequency order. The method initiates discontinuities in the signal of the chirp because of the mismatch in the segments of each phase. A similar experiment using overlap and window sizes draws the same observations.

In measuring the pitch period and formants of a speech segment, drastic changes in the format frequencies and in the voicing are observed. In this observation of the slices of speech, it is realistic to expect some reduction in errors by a factor of ten in the format frequency dimensions within the dimension above 1.5F0 (Eriksson & Traunmiller, 1974).

## References

Eriksson, A. & Traunmiller, H. (1974). A Method of Measuring Format Frequencies at High Fundamental Frequencies. http://www2.ling.su.se/staff/hartmut/a0333.pdfFessler, J. (2004). The Discrete Fourier Transform. http://web.eecs.umich.edu/~fessler/course/451/l/pdf/c5.pdf

Ondrej, F. & Martin, C. (2006). FFT and the Convolution Performance in Image Filtering on GPU. Department of Computer Science and Engineering, Czech Technical University in Prague. http://cadik.posvete.cz/papers/cadikm-iv06-gpu.pdfManolaskis, D. G. & Proakis G. J. (1996). Digital Signal Processing: Principles, Algorithms, and Applications. Third Edition. http://d1.amobbs.com/bbs_upload782111/files_17/ourdev_468290.pdf

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