Mastering The Fourier Transform: From Theory To Application PDF

MASTERING THE FOURIER TRANSFORM: FROM THEORY TO APPLICATION FIDEL CERTUCHE  INTRODUCTION TO FOURIER TRANSFORM  Historical background: The Fourier Transform is named after Jean-Baptiste Jo...

MASTERING THE FOURIER TRANSFORM: FROM THEORY TO APPLICATION FIDEL CERTUCHE  INTRODUCTION TO FOURIER TRANSFORM  Historical background: The Fourier Transform is named after Jean-Baptiste Joseph Fourier (1768–1830), a French mathematician and physicist. Fourier introduced the concept that any periodic/no periodic function could be represented as a series of sines and cosines, which is the foundation of the Fourier series, in his seminar work "Théorie analytique de la chaleur" ("The Analytic Theory of Heat") published in 1822(brief explanation of this experiment). Fourier Transform Discrete Fourier Transform  Definition: The FT converts a time-domain signal, into a frequency-domain representation. This transformation is essential because, in many instances, analyzing the signal in the frequency domain offers insights that are not readily apparent in the time domain. (examples will be given in this part that support this paragraph statement) The Continuous Fourier Transform of a time-domain signal f(t) is given by: F(ω) is the Fourier Transform of f(t), t represents time, ω is the angular frequency in radians per second, e^−iωt is the complex exponential function, i is the imaginary unit SQR(−1).  Decomposition of Fourier Transform Equation Euler Complex Exponential equation magnitude Here it will be explained relationship between Fourier equation and triangle, phase, magnitude, catheti. Matlab example (next slide GUI of the app) Phase angle The triangle formed by the real and imaginary parts of a complex exponential (and by extension, any complex number resulting from the transform) is not just a geometric figure but a representation of the magnitude and phase of a sinusoidal component in the signal. The legs of the triangle (catheti) represent the real and imaginary components, and the hypotenuse represents the magnitude of the frequency component. The angle formed by the hypotenuse with the real axis represents the phase shift of the component.  This means that when applying either CFT or DFT for doing the Fourier analysis, it will always be possible to reconstruct the signal by applying the inverse CFT or DFT. Source: Intuitive Guide to Fourier Analysis and Spectral Estimation with MATLAB by Charan Langton Matlab example of fourier synthesis (next slide GUI of the app)  DFT - FFT From now on we are going to the digital domain, and the fourier analysis (DFT and FFT) will be applied to sampled signals Source: W. Scherr, FH Digital IC Design 1 course  DFT The DFT transforms a finite sequence of equally-spaced samples of a function (often a time-domain signal) into a same-length sequence of equally- spaced samples in the frequency domain. ⚫ X[k] is the k-th element of the frequency domain sequence The number of samples in the time domain is usually represented by the variable N. While N can be any positive integer, a power of two is usually chosen, i.e. 128, 256, 512, 1024, etc. There are two reasons for this. First, digital data storage uses binary addressing, making powers of two a natural signal length. Second, the most efficient algorithm for calculating the DFT, the Fast Fourier Transform (FFT), usually operates with N that is a power of two. Typically, N is selected between 32 and 4096. In most cases, the samples run from 0 to N-1 , rather than 1 to N (W. Smith, 1999, p. 161).  COMPUTATION OF A DFT  Explanation of terms: frequency bins, phase, amplitude  Simple numerical example (Python )  FFT Algorithm used by numpy´s python library to compute DFT Cooley Tukey Algorithm Divide-and-conquer approach to computing the Discrete Fourier Transform (DFT) of a composite size N= n1 * n2. The original DFT is broken down into smaller DFTs, which are then solved recursively. This process continues until the transform is reduced to trivial computations, typically DFT´s of size 1 or 2. The Cooley-Tukey FFT algorithm leverages two main strategies called Decimation in Time (DIT) and Decimation in Frequency (DIF) to efficiently compute the Discrete Fourier Transform (DFT). These strategies manage the order in which input data is processed and how the DFT is computed and recombined. DIT: The sequence is recursively divided based on even and odd indexed elements before combining. DIF: The sequence is divided after applying the DFT, focusing on separating the output into its component parts  Decimation in Time The FFT algorithm splits the sequence of input data into smaller parts based on even and odd indices. This process is recursively repeated, reducing the size of the problem at each step. The input sequence is divided into two halves: one containing the elements at even indices and the other containing elements at odd indices. For a sequence of: x,x,x,...,x[N−1], where N is the total number of samples, the division results in: Even indexed sequence: x,x,x,...,x[N−2] Odd indexed sequence: x,x,x,...,x[N−1] Apply the FFT separately on the even and odd indexed sequences. Since each sequence is now half the size of the original, the complexity of the problem reduces significantly. After computing the FFTs of the two halves, the results are combined using the butterfly operations. The combination uses the formula:. Twiddle factors, what are these for? where E[k] and O[k] are the FFTs of the even and odd indexed parts, respectively  Decimation in Frequency Decimation in Frequency, on the other hand, approaches the problem by initially calculating the full DFT and then dividing the output frequency spectrum into smaller components. Instead of splitting the input sequence by indices, the FFT is first applied to the whole sequence to get a full spectrum. This spectrum is then split into lower and upper halves, which represent lower and higher sets of frequencies. Recursively apply FFTs on the separated frequency halves. This method simplifies the DFT calculations as it reduces the frequency resolution at each recursive step, dealing progressively with smaller and smaller frequency bins. The lower and higher frequency components are recursively split and processed until the transform operations reach the base case (typically DFTs of size 1 or 2). The results are then recombined using butterfly operations. Both concepts will be useful when explaining FFT HW impementation with butterfly structure.  Conclusion on Both Decimation Methods: Data Handing, Operational Flow, Butterfly Application DIT Data Handling: DIT splits the sequence based on time indices—specifically, it separates the input data into even and odd indexed elements. This process begins with the full sequence and repeatedly divides it into smaller subsequences until it reaches the smallest possible divisions. Operational Flow: The FFT is applied first to these smaller subsequences, and the results are combined using the butterfly operations. The combination of these FFT results from the ground up forms the final FFT of the original sequence. Butterfly Application: In DIT, the butterfly operations combine the results of the FFTs applied to these subsequences. These operations involve basic arithmetic and the use of twiddle factors to adjust the phase of the combined outputs, gradually building up to the full FFT. DIF Data Handling: DIF divides the sequence based on frequency components—it starts with the full FFT spectrum and works its way down to individual components by splitting the frequency spectrum into halves. This approach essentially reverses the operational flow compared to DIT. Operational Flow: The initial FFT computations produce a broad overview of the frequency spectrum, which is then successively divided into smaller and more detailed components. The final stage of the DIF approach ends with the detailed components being directly related to the time-domain samples. Butterfly Application: In DIF, butterfly operations are used to refine the frequency components, with each step dividing the spectrum into increasingly finer details. Twiddle factors are applied to adjust phases as the algorithm progresses from the complete spectrum to individual components. Simple FFT Calculation Examples (Author: W. Scherr) Jupyter Notebook  INTRODUCTION TO FAST FOURIER TRANSFORM DFT is slower and not very practical for large datasets, so here is when FFT steps in as a set of algorithms to compute the DFT in a more efficient manner, exploiting symmetries and redundancies in the DFT's computation. It reduces the computational complexity significantly: Cooley-Tukey Algorithm Radix-2 FFT Algorithm (powers of 2) Radix-4 FFT Algorithm (powers of 4) Split-Radix FFT Algorithm Mixed-Radix FFT Algorithm Prime Factor Algorithm (PFA) Bluestein's FFT Algorithm FFT algorithms typically use a divide-and-conquer approach, breaking the DFT into smaller DFTs recursively, which helps reduce the number of computations needed. The most common FFT algorithm is the Cooley-Tukey algorithm, which is efficient when N is a power of two.  COMPUTATION OF A FFT FOR DIGITAL IMPLEMENTATION  Simple numerical example (Python ). Timings will be compared for a DFT and FFT calculation over a random signal.  HW IMPLEMENTATIONS The software version of the FFT has, as limitation, the execution of instructions serially (one at a time) and is therefore severely constrained by the processor instruction throughput. The hardware FFT performs many of its processing tasks in parallel, hence can achieve order-of-magnitude throughput improvements over software FFTs executed in DSP microprocessors. Pipelined architectures in the hardware implementation of Fast Fourier Transform (FFT) are designed to enhance the speed of FFT calculations, which are crucial for a variety of applications like digital signal processing, communications, and image processing. The concept behind is to utilize a sequence of processing elements arranged such that multiple stages of the FFT calculation can be performed simultaneously, or "in pipeline". What are the constraints and challenges when implementing FFT HW processors? Area Requirements and Power Consumption  CASE STUDIES AND APPLICATIONS IN TELECOMMUNICATIONS AND ACOUSTICS Multicarrier modulation techniques, such as orthogonal frequency-division multiplex (OFDM) and discrete multitone (DMT), have received great attention in high-speed data communication systems (multicarrier modulation based transceivers involve fast Fourier transform (FFT) computations that require a large amount of arithmetic operations) and have been selected for several communication standards, such as wireless local area network (WLAN), asymmetric digital subscriber line (ADSL), very high-speed digital subscriber line (VDSL),digital audio broadcasting (DAB), digital video broadcasting (DVB), powerline communications (PLC). Challenge: Pipeline architectures use high areas on silicon but are fast. The shared memory architecture is area efficcient but cannot get high speeds unless higher radix algorithm is used. Memory architectures using radix-2 requires 4 times more computational cycles than radix-4 algorithm. But radix-4 cannot process 128, 512, 2048 or 8192 p0oints FFT cause are not powers of 4 Proposal: Continuous-Flow Mixed-Radix (CFMR) FFT Processor The MR algorithm uses both the radix-4 and radix-2 and can perform fast FFT computations and can process FFTs that are not powers of four. The proposed continuous-flow MR (CFMR) FFT processor using the MR algorithm can perform FFT computations regardless of the length of FFT. What are Radix-2 and Radix-4 algorithms?  APPLICATION IN ACOUSTICS (CONT) Time to Frequency Domain Conversion: The primary function of an FFT analyzer is to take a time-domain signal (like a recording of audio) and convert it into its frequency-domain representation. This allows users to see which frequencies are present in the signal and at what intensities. Resolution: FFT analyzers provide a spectrum view where signals are represented in terms of their constituent frequencies. The resolution of FFT Analyzer this spectrum can vary based on the settings, especially the number of FFT points or “lines”. Filtering and Weighting: Most FFT analyzers allow users to apply different filters and weightings to the input signal. This is especially relevant in applications like audio analysis, where the A, B, C, or Z-weighting filters can be applied to mimic the human ear’s response to different frequencies. Windowing: To mitigate the effects of processing finite chunks of data (which can introduce artifacts into the frequency analysis), FFT analyzers use various window functions like Hanning, Rectangle, Flat Top, or Kaiser-Bessel. Applications: FFT analyzers are utilized in a variety of fields, including acoustics (to analyze sound or noise signals), vibration analysis (to determine the frequencies at which structures or machines might resonate), and telecommunications (to analyze the frequency content of signals). Source: www.svantek.com  EMERGING TECHNOLOGIES BASED ON FFT 1. Quantum Computing 2. 5G and Beyond Wireless Networks 3. Advanced Radar and Satellite Imaging 4. Deep Learning and Neural Networks 5. Bioinformatics and Health Diagnostics 6. Environmental Modeling and Climate Science  BIBLIOGRAPHICAL REFERENCES Bracewell, R. N. (2000). The Fourier Transform and Its Applications. McGraw-Hill. Oppenheim, A. V., & Schafer, R. W. (2009). Discrete-Time Signal Processing. Prentice Hall. Smith, S. W. (1997). The Scientist and Engineer's Guide to Digital Signal Processing. California Technical Publishing. Stein, E. M., & Shakarchi, R. (2003). Fourier Analysis: An Introduction. Princeton University Press. Proakis, J. G., & Manolakis, D. G. (2006). Digital Signal Processing: Principles, Algorithms, and Applications (4th ed.). Pearson. Lyons, R. G. (2010). Understanding Digital Signal Processing (3rd ed.). Pearson. Poularikas, A. D. (2015). The Transforms and Applications Handbook (3rd ed.). CRC Press. Bracewell, R. N. (1999). Fourier Analysis and Imaging. Springer.  Smith, J. O. (1997). Mathematics of the Discrete Fourier Transform (DFT) with Audio Applications. W3K Publishing. Charan, L. The Intuitive Guide to Fourier Analysis and Spectral Estimation with Matlab. Oppenheim, A. V., Willsky, A. S., & Hamid, S. (1996). Signals and Systems. Prentice Hall Georgia Institute of Technology. (n.d.). CSpin. Retrieved May 1, 2024, from https://dspfirst.gatech.edu/matlab/ZipFiles/cspin Georgia Institute of Technology. (n.d.). Fseriesdemo. Retrieved May 1, 2024, from https://dspfirst.gatech.edu/matlab/ZipFiles/fseriesdemo Slade, G. W. The Fast Fourier Transform in Hardware: A Tutorial Based on an FPGA Implementation. Mario Garrido, A Survey on Pipelined FFT Hardware Architectures, Journal of Signal processing Systems Byung G. Jo and Myung H, New Continuous-Flow Mixed-Radix (CFMR) FFT Processor Using Novel In-Place Strategy, IEEE Transactions on Circuits and Systems Lihong Jia, Yonghorig Gao , A Pipelined Shared-Memoyy Architecture for FFT Processors, IEEE Explore W. Scherr, simple FFT calculation examples , Carinthia University of Applied Sciences