dft - ECE 421 Sum 2010 Notes Set 11 Discrete Fourier...

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Unformatted text preview: ECE 421 - Sum 2010 Notes Set 11 : Discrete Fourier Transforms 1 INTRODUCTION The frequency domain characteristics of signals and systems are very important properties in signal processing, communications systems, circuit design, and other areas. In design and analysis, we frequently use analytical methods, such as the Laplace and Fourier trans- forms, to determine theoretical frequency domain properties. However, a new requirement arises when we choose to implement a system using digital signal processing operating on real-world sampled signals. These signals are produced by information processes and/or exist in noisy environments, and rarely have an analytical form or mathematical function form. We are therefore led to the following question: How do we determine the frequency-domain properties of a sampled signal, acquired from real-world processes, which does not have an explicit analytical form? For one example of the above situation, how would we determine the frequency response of a system whose impulse response was obtained by sampled measurements? As another example, how would we obtain the spectrum of a digitized audio or video information signal, which are inherently non-predictable? To answer these and other questions, digital signal processing has developed computa- tional methods for obtaining the frequency content of such signals. One method is called the Discrete Fourier Transform (DFT) and the computational algorithm which frequently implements the DFT is called the Fast Fourier Transform, or FFT. The DFT/FFT is a widely used method in many applications of Digital Signal Processing. In this set of Notes we will study the basics of the DFT. These basic properties and tech- niques we learn will be valid for any application of the DFT. In a future set of Notes we will learn about the Inverse DFT, which has turned out to be very important in many high-speed internet access systems. ECE 421 - Sum 2010 Notes Set 11 : Discrete Fourier Transforms 2 RELATION TO CONTINUOUS TRANSFORM One important use of the DFT is to numerically compute the spectrum of sampled signals. Recall that sampling a continuous time-domain waveform x t with a time-domain sam- pling interval T s produces the digital signal x [ m ℄= x mT s . Taking the DFT of x [ m ℄ then allows us to compute the spectrum of the discrete-time signal. The DFT used in this man- ner is sometimes called a DFT Spectrum Analyzer (or Digital Spectrum Analyzer) and the technique is called DFT Spectrum Analysis. In this introduction, we will assume infinite precison on the sampled time-domain signal. To develop this approach, let x t be a continuous analog signal defined over 0 t T and let X f be its continuous Fourier transform: X f = F f x t g = Z T t = x t e j 2 π f t dt 1 Let the time-domain sampling interval be T s , and the number of time domain samples be M , such that T = M T s 2 Now replace continuous dt , t , and f in (1) with their sampled approximations dt T s ; t mT s ; f k...
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dft - ECE 421 Sum 2010 Notes Set 11 Discrete Fourier...

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