##### Sinc Function, Ssb-Am, Fm

FM Name: Onur Mustafa Erdogan ID Number: 10014044 Submission Date: 24. 12. 2012 Abstract: In these project, I will analyze Fourier Transform of sinc function and it’s modulation. (SSB-AM,FM) I will explain SSB-AM and FM theoretically and solve their math model in steps. After all, I’ll use simulations and graphics to prove my solutions and In the and, I will write my conclusions down. Introduction SSB-AM In DSB-SC it is observed that there is symmetry in the band structure. So, even if one half is transmitted, the other half can be recovered at the received.

By doing so, the bandwidth and power of transmission is reduced by half. Depending on which half of DSB-SC signal is transmitted, there are two types of SSB modulation; 1. Lower Side Band (LSB) Modulation 2. Upper Side Band (USB) Modulation FM Frequency modulation (FM) is a method of impressing data onto an alternating-current (AC) wave by varying the instantaneous frequency of the wave. This scheme can be used with analog or digital data. Frequency modulation is similar in practice to phase modulation (PM). When the instantaneous frequency of a carrier is varied, the instantaneous phase changes as well.

The converse also holds: When the instantaneous phase is varied, the instantaneous frequency changes. But FM and PM are not exactly equivalent, especially in analog applications. When an FM receiver is used to demodulate a PM signal, or when an FM signal is intercepted by a receiver designed for PM, the audio is distorted. This is because the relationship between frequency and phase variations is not linear; that is, frequency and phase do not vary in direct proportion. System Model We are looking for Fourier Transform of sinc function, in first step, we put our function in Fourier Transform Formula. if and if  and

After these equations, we can say that Fourier Transform of sinc(t)=? rect(w/2) Next step is modulating our signal for transmission. I will use SSB-AM and FM for this. SSB-AM for x(t)=sinc(t); General Formula for SSB-AM is; Upper sideband Lower sideband H{x(t)} is Hilbert Transform, it’s defined as; I put sinc(t) in the formula and used Hilbert Transform; Upper Sideband Lower Sideband And in the frequency domain I reached; Upper Sideband Lower Sideband FM for x(t)=sinc(t) We can show FM signal as; But we need to know phase; Let’s put sinc(t) instead of x(t)  We can use Dirichlet Condition to solve this integral.

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