The Fourier Transform

The Fourier series is an effective tool for representing periodic functions like a train of square waves. Consider a pulse-train waveform consisting of a …

Among the key elements of the Fourier Transform, the sinc function is unique in that it equals 1 when its argument is zero and exhibits even symmetry. In …

In radio broadcasting, Fourier Transform properties are applied in simultaneous multi-channel transmission, adjusting audio clip speeds, live broadcast …

The Frequency Shifting property of Fourier Transforms states that a shift in the frequency domain corresponds to a phase shift in the time domain, …

Parseval's theorem is a principle used in signal processing to calculate the energy of a signal. It allows the computation of the same energy value …

The Discrete-Time Fourier Transform is a variant of the Fourier transform applied to a discrete-time signal. This transform replaces the integral in the …

Consider two discrete-time signals, each with their respective Discrete-Time Fourier Transforms DTFTs. The signals are first multiplied by constants a and …

Consider a discrete-time Fourier transform (DTFT) pair, differentiate both sides with respect to Ω, and then multiply by j. The right-hand side …

Consider a vibration sensor that continuously captures data in the form of a continuous time-dependent signal. However, in reality, the sensor can only …

The Fast Fourier Transform, FFT, is a computational algorithm for calculating the Discrete Fourier Transform by breaking the calculations into smaller, …