[Iqbal et. al., Vol.7 (Iss.1): January 2020] ISSN: 2454-1907 DOI: 10.29121/ijetmr.v7.i1.2020.495 Http://www.ijetmr.com©International Journal of Engineering Technologies and Management Research [36] THEKERNEL OF N- DIMENSIONAL FRACTIONAL FOURIER TRANSFORM Zakia Abdul Wahid 1 , Saleem Iqbal *1 , Farhana Sarwar 2 , Abdul Rehman 1 1 Department of Mathematics, University of Balochistan Quetta, Pakistan 2 Department of Mathematics F.G. Girls Degree College, Madrissa Road, Quetta, Cantt, Pakistan Abstract: In this paper we have developed the kernel of N-dimensional fractional Fourier transform by extending the definition of first dimensional fractional Fourier transform. The properties of kernel up to N- dimensional are also presented here which is missing in the literature of fractional Fourier transform. The properties of kernel of fractional Fourier transforms up to N- dimensional will help the researcher to extend their research in this aspect. Keywords: Fourier Transform; Fractional Fourier Transform; N-Dimensional Fractional Fourier Transform; Kernel of N- Dimensional Fractional Fourier Transform. Cite This Article: Zakia Abdul Wahid, Saleem Iqbal, Farhana Sarwar, and Abdul Rehman. (2020). “THEKERNEL OF N- DIMENSIONAL FRACTIONAL FOURIER TRANSFORM.” International Journal of Engineering Technologies and Management Research, 7(1), 36-41. DOI: 10.29121/ijetmr.v7.i1.2020.495. 1. Introduction The idea of fractional operator of Fourier transform (FT) was introduced by V. Namias in 1980 [4]. In which he had descripted first time the comprehensive definition and mathematical frame work of Fractional Fourier Transform (FRFT). The Fractional Fourier transform (FRFT) depends on a parameter  that is associated with the angle in phase plane. This leads to the generalization of notion of space (or time) and frequency domain which are central concepts of signal processing. The kernel of the fractional FT is except for a phase factor, equal to the propagator of the non-stationary Schrodinger equation for the harmonic Oscillator, this transform is also used in optics [10,11]. FRFT was first introduced as a way to solve certain classes of ordinary and partial differential equations arising in quantum mechanics [4]. FRFT has found applications in areas of signal processing such as repeated altering, fractional convolution and correlation, beam forming, optional filter, convolution, filtering and wavelet transforms, time frequency representation [7]. The FRFT is basically a time-frequency distribution. It provides us with an additional degree of freedom (order of the transform), which is in most cases results in significant gain over the classical Fourier transform. With the development of FRFT and related concepts, we see that the ordinary frequency domain is merely a special case of a continuum of fractional Fourier domains. Every property and application of the ordinary Fourier transform becomes a special case of the FRFT. So, in every area in which Fourier