Digital Signal Processing - Fourier Transform PDF

DIGITAL SIGNAL PROCESSING – FOURIER TRANSFORM  Course Learning Outcomes Course Objectives Course Learning Outcomes 1. Explain the advantages 1. Grasp the advantages and and applications of digital applications of digital signal signal processing. p...

DIGITAL SIGNAL PROCESSING – FOURIER TRANSFORM  Course Learning Outcomes Course Objectives Course Learning Outcomes 1. Explain the advantages 1. Grasp the advantages and and applications of digital applications of digital signal signal processing. processing and handle discrete time processing of continuous time signals Contents Fourier Series Fourier Transform Discrete-Time Fourier Series (DTFS) Discrete-Time Fourier Transform (DTFT) Discrete Fourier Transform (DFT) Similarities, Differences, and Applications A voice recording or any natural sound is typically aperiodic because the sound patterns are not repetitive. A noise signal Key Points Repetition: Periodic signals repeat over time, while aperiodic signals do not. Frequency Domain: Periodic signals have discrete frequency components, while aperiodic signals have continuous spectra. Fourier Series. What is Fourier Transform? Definition: Converts a time-domain signal into its frequency-domain representation. Decomposes a signal into sinusoids of different frequencies. Continuous and applies to analog signals. Works on aperiodic signals. Provides continuous frequency spectrum Applications: Used in audio processing, image compression, and communication systems. Discrete-Time Fourier Series (DTFS) Definition: Fourier Series applied to periodic discrete-time signals. Discrete periodic signals. N-point periodicity. Discrete-Time Fourier Transform (DTFT) Extends Fourier Transform to discrete-time signals. Continuous frequency domain, applied to non-periodic discrete-time signals. Used for theoretical analysis in DSP. Discrete Fourier Transform (DFT) Converts a finite sequence of samples into a discrete frequency-domain representation. Finite-length sequences. Output is a discrete set of frequency components. Comparison: DTFT vs. DFT Comparison: Fourier Series vs. Fourier Transform Inverse Discrete Fourier Transform (IDFT) Converts the frequency-domain representation back to the time- domain. Practical Applications of Fourier Analysis Signal filtering in audio and communications. Data compression (JPEG, MP3). Spectrum analysis in radar and wireless communications. Example: Using DFT to reduce noise in a speech signal. DFT-Discrete Fourier Transform (In detail) EXAMPLE  Solution (a) Given sequence is x(n) = {1, –1, 2, –2}. Here the DFT X(k) to be found is N =4-point and length of the sequence L = 4. So no padding of zeros is required. We know that the DFT {x(n)} is given by Example 2 Compute the DFT of the 3-point sequence x(n) = {2, 1, 2}. THANK YOU