Prove that if F(ω) is the Fourier transform of f(t) ℱ[f(t) sinω0 t]=(j)/(2)[F(ω+ω0)-F(ω-ω0)]

Name: Problem 16, Chapter 17: Fundamentals of Electric Circuits
Uploaded: 2022-03-03T05:03:54-08:00
Duration: 1 min 35 s
Channel: Amit Srivastava
Description: Prove that if F(ω) is the Fourier transform of f(t) ℱ[f(t) sinω0 t]=(j)/(2)[F(ω+ω0)-F(ω-ω0)]

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Prove that if F(ω) is the Fourier transform of f(t) ℱ[f(t)...

Prove that if $F(\omega)$ is the Fourier transform of $f(t)$
$$\mathcal{F}\left[f(t) \sin \omega_{0} t\right]=\frac{j}{2}\left[F\left(\omega+\omega_{0}\right)-F\left(\omega-\omega_{0}\right)\right]$$

Key Concepts

Fourier Transform

The Fourier Transform is a mathematical tool that decomposes a function of time (or space) into its constituent frequencies. It plays a central role in analyzing systems behavior in the frequency domain and establishes a correspondence between the time domain and the frequency domain representations of a signal.

Modulation (Multiplication by a Sinusoid)

Modulation refers to the process of multiplying a time-domain signal by a sinusoidal function. This operation shifts the frequency content of the original signal, which is a fundamental technique in communications and signal processing for tasks like frequency translation.

Frequency Shifting Property

The frequency shifting property of the Fourier Transform states that multiplying a time-domain signal by an exponential function results in a frequency-domain shift of the signal's Fourier transform. When the modulation involves sine or cosine functions, which can be expressed in terms of exponentials via Euler’s formula, the Fourier transform is shifted to new frequency bands corresponding to the modulation frequency.

Euler’s Formula

Euler’s Formula provides a relationship between complex exponentials and sinusoidal functions. By expressing sine and cosine in terms of complex exponentials, it becomes easier to apply the Fourier Transform's properties, such as linearity and frequency shifting, to signals that are modulated by sinusoidal functions.

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