# Representing Fourier Transform of Periodic Signal in LTI System

• SrEstroncio
In summary, the Fourier transform of a periodic signal, represented by a sequence of triangle pulses symmetric to the origin, can be reduced to constants by using the formula 2\pi (sum (Xn delta(omega - (n)omega0)). To obtain the Fourier transform of a triangle pulse, you can use the transform of a rectangular pulse and the convolution theorem, as the convolution of two rectangular pulses is a triangular pulse.
SrEstroncio
How do you represent the Fourier transform of a periodic signal as the input in a LTI system?
More precisely, a sequence of triangle pulses symmetric to the origin.

I know what the Fourier transform is 2\pi (sum (Xn delta(omega - (n)omega0)))

I was told to reduce the Xn to constants, but can't figure how to reduce the n omega0.

This may not help you, but you may obtain the Fourier transform of a triangle pulse using the transform of a rectangular pulse and the convolution theorem. The convolution of two rectangular pulses is a triangular pulse.

$X(n)=\frac{1}{T}\int_{t_0}^{t_0+T} f(n)e^{-i2\pi nt/T}$

## 1. What is the Fourier Transform of a periodic signal?

The Fourier Transform of a periodic signal is a mathematical representation of the signal in the frequency domain. It decomposes the signal into its individual frequency components, showing the amplitude and phase of each component.

## 2. How is the Fourier Transform of a periodic signal different from a non-periodic signal?

The Fourier Transform of a periodic signal results in a discrete spectrum with only specific frequency components, while the Fourier Transform of a non-periodic signal results in a continuous spectrum with a range of frequency components.

## 3. How is the Fourier Transform of a periodic signal represented in an LTI system?

The Fourier Transform of a periodic signal can be represented in an LTI (Linear Time-Invariant) system using the transfer function, which is a mathematical relationship between the input and output signals. The transfer function is the Fourier Transform of the impulse response of the LTI system.

## 4. Can the Fourier Transform of a periodic signal be used to analyze the behavior of an LTI system?

Yes, the Fourier Transform of a periodic signal can be used to analyze the behavior of an LTI system. By examining the magnitude and phase of the transfer function at different frequencies, we can determine how the system responds to different input frequencies.

## 5. What is the relationship between the Fourier Transform of a periodic signal and the frequency response of an LTI system?

The frequency response of an LTI system is the magnitude and phase response of the system to different input frequencies. It can be obtained by evaluating the transfer function of the system at different frequencies, which is essentially the Fourier Transform of the periodic signal. Therefore, the Fourier Transform of a periodic signal is directly related to the frequency response of an LTI system.

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