Digital Signal Processing Lab: Circular Convolution, Fast Fourier Transform and IIR Filter
Digital Signal Processing Lab 2023 Spring Victory won’t come to us unless we go to it. Contents Lab 1 Digital Signal Processing (1) 1 1.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.3 Theory Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Lab 2 Digital Signal Processing (2) 3 2.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Lab 3 Additional T opics (Optional) 4 3.1 T opics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Lab 1 Digital Signal Processing (1) 1.1 Objectives Understand the theoretical concepts of circular convolution. Learn to implement circular convolution in MATLAB. 1.2 Problem Statement The primary aim of this lab exercise is to gain a practical understanding of circular convolution in MAT - LAB. 1.3 Procedure Enter the sequence x[n]; Enter the sequence y[n]; Find the lengths of x[n]andy[n]ie;NxandNyrespectively Check Nx=Ny: proceed if equal Initialize a loop variable number of output points For each out sample, access the samples of y[n]in the cyclic order Find the sum of products of x[n]and cyclically folded and shifted y[n] 1.4 Experiments Implementation of circular convolution in MATLAB. Lab 2 Digital Signal Processing (2) 2.1 Objectives Understand the theoretical concepts of DFT and FFT . Learn to implement DFT and FFT in MATLAB. Analyze and interpret signals in the frequency domain using DFT and FFT . Understand and observe the computational advantages of FFT over DFT . Learn to use MATLAB’s built-in functions for DFT and FFT . 2.2 Problem Statement The primary aim of this lab exercise is to gain a practical understanding of DFT and FFT in MATLAB. W e will implement these transforms from scratch and compare the results and performance with MATLAB’s in-built functions. We will further analyze the effect of windowing and zero-padding on these transformations. 2.3 Theory Background The Fourier Transform is a fundamental concept in signal processing. It allows us to transform a signal from its original time or spatial domain into a representation in the frequency domain and vice versa. The Discrete Fourier Transform (DFT) is a version of the Fourier Transform that works with a finite, discrete number of data points. The Fast Fourier Transform (FFT) is an algorithm for computing the DFT in a more efficient way . FFT algorithms are so commonly employed to compute DFT s that the term ’FFT’ is often used to mean ’DFT’ in colloquial settings. Windowing is a technique used in signal processing to manage spectral leakage, a phenomenon that occurs when a DFT is performed on a non-periodic signal. Zero-padding is a technique where we append zeros at the end of the data sequence. This increases the number of DFT points and results in a smoother and more continuous Fourier Transform, which makes it easier to interpret. 2.4 Experiments Implementation of DFT and FFT . 2.4.1 Implement the Discrete Fourier Transform and Fast Fourier Transform manually in MATLAB Use a simple sinusoidal signal x[n] = 2 ∗sin(2000 πnt s)as your test signal Plot both the time-domain and frequency-domain representation of the signal. 2.4 Experiments 2.4.2 Comparison with Built-in Functions Compare your results of DFT and FFT with MATLAB’s built-in ‘fft’ function using different signals. Use signals of varying complexities (e.g., a single sinusoid, multiple sinusoids, a square wave, etc.). 2.4.3 Performance Comparison between DFT and FFT Compare the computation time between your DFT and FFT implementation. Analyze why the FFT is faster and how the speedup scales with the size of the input. 2.4.4 Effect of Windowing on DFT and FFT Investigate the effects of windowing on the spectral leakage of signals. This includes implementing different windows (Rectangular, Hamming, Hanning, etc.), applying them to signals, and analyzing the results in the frequency domain. 2.4.5 Effect of Zero-Padding in DFT and FFT Explore the effects of zero-padding on the resolution of DFT and FFT . Zero-pad signals to different lengths and analyze how it affects the frequency domain representation of the signal. 3 Lab 3 Digital Signal Processing (3) 3.1 Objectives Design and implementation of IIR filter to meet given specifications. 3.2 Procedures Get the pass band and stop band edge frequencies Get the pass band and stop band ripples Get the sampling frequency Get the sampling frequency Find the filter coefficients Plot the magnitude response 3.3 Experiments Implement IIR filter to meet given specifications.
