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# SIGNALSANDSYSTEMSANALYSIS ECE1552

Pitt

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This 5 page Class Notes was uploaded by Macy Lowe on Monday October 26, 2015. The Class Notes belongs to ECE1552 at University of Pittsburgh taught by Staff in Fall. Since its upload, it has received 41 views. For similar materials see /class/229456/ece1552-university-of-pittsburgh in Electronics and Computer Technology at University of Pittsburgh.

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Date Created: 10/26/15

Notes on the discrete Fourier transform JR Boston January 2006 page 1 Notes on the discrete Fourier transform These notes provide a brief overview of the discrete Fourier transform DFT The DFT is used in computer processing of discrete signals in the same manner and for many of the same purposes as the continuous Fourier transform for continuous signals that you met in Signals and Systems Analysis ECE 1552 These purposes include such tasks as describing how the energy or power of a signal is distributed over frequency and determining the output of a system for a given input A more formal treatment can be found in most Signals and Systems texts the notation used here is based on Jackson Signals Systems and Transforms AddisonWesley 1991 A fundamental difference between processing continuous signals and processing discrete signals with a computer is that the computer can only represent a nite number of samples of a signal Hence a continuous signal must be sampled at discrete time points de ned by a sampling rate and only a nite length or duration of the signal can be analyzed These two limitations have important implications for the interpretation of the results of Fourier analysis as will be shown below The Direct Transform Assume that we have a possibly in nite sequence xn with discretetime Fourier transform DTFT Xejm where Xejm Z xnejDn D is a continuous variable Remember that the nm DTFT is periodic in D with period 27E This sequence is often generated by sampling a function of time but it may arise directly For example the consumer price index is a discrete time sequence that is only calculated once per month In a practical computerbased signal processing situation we can only deal with a nite segment of xn say a sequence xWn ofN points from n 0 to n Nl We can relate this nite signal to xn by realizing that xWn is the product of xn and a rectangular window wn of length N which will be referred to as rectWn Consider the sampled sequence xWn of length N obtained from the in nite signal xn xn 0 S n S Nl lenl 0 otherwise xWn rectN n xn where l 0 S n S Nl rectN n 0 otherwise Notes on the discrete Fourier transform JR Boston January 2006 page 2 The discretetime Fourier transform DTFT of XWII is its ztransform evaluated on the unit circle that is N l N l DTFNXWMZXWnZjm ZXW meal ZXW mew n0 0 where D is a continuous variable discrete frequency The difference between the DTFT of XIl and the DTFT of XWII is that the latter involves a nite sum Since 0 is continuous we cannot represent it directly in the computer Instead we can only represented a sampled version The discrete Fourier transform DFT is just that a sampled version of the discretetime Fourier transform Xwm where the samples are at N equally spaced frequencies 0 27EkN k 0 l 2 Nl on the unit circle e39u The direct DFT is de ned by N l 2 W zxw meJZMMN Xwk fork012N1 CD 7 n0 N l DFTXWn ZXW newn n0 Since the DFT is de ned by a nite set of algebraic equations there is no issue of the region of convergence It is always de ned for Xw Note that both Xwn and XW k are just sequences of numbers in MATLAB Which is frequency and which is time has to be interpreted by the programmer As is the case for continuous Fourier transforms multiplication in one domain of the DFT implies convolution in the other Since XWII is the original signal multiplied by a rectangular window function the DFT of XW n is the DFT of the original signal convolved with the DFT of a rectangular window which is a sinc function This convolution tends to smear the spectrum of the original signal See page 12 of the Notes on Spectral Estimation for more details on the effects of windows The de nition of the DFT differs from the discrete Fourier series DFS in that the DFS coefficients are de ned by a sum that is divided by N while the DFT is de ned by an un normalized sum This difference parallels the difference between the continuous time Fourier series and Fourier transform and is due to the fact that the continuous and discrete Fourier series describe nite power signals in nite duration periodic signals and the continuous and discrete time Fourier transforms describe nite energy signals nite duration signals Resolution of the DFT How do we interpret the indices k 0 l 2 Nl Both the sequence XWII and the DFT Xwk are represented in the computer as vectors We often interpret the sequence Xwn as a time series that was sampled at a frequency of fS Hz where the increment from n to n1 represents the sampling interval T lfs In that case the total length in seconds of the signal being analyzed is NT N samples of Xwk represent one period of discrete frequency 27 radians Therefore the discrete frequency increment from k to kl is Am 27tN This increment can be interpreted in Notes on the discrete Fourier transform JR Boston January 2006 page 3 terms of the original sampling frequency by noting that discrete frequency 03 2n corresponds to the sampling frequency f5 Hence Am 27tN corresponds to Af fsN where Am is the increment in discrete frequency in radians and Af is the related increment in real frequency in Hz or cycles per second The length of the DFT in terms of real frequency is NAf f5 Actually the samples in the DFT go from 0 Hz to fS Af since there are only N samples and the rst sample is at 0 Hz Note that the discrete frequencies from TE to 271 real frequencies fS 2 to f5 actually correspond to the negative frequencies of the Fourier transform For example if we sample a 500 Hz sine wave at fS 8000 Hz for 05 seconds we have a data record 4000 points long N 4000 The DFT will include 4000 points with Am 000057 radians In terms of the original frequencies we would say that the DFT has a frequency resolution Af of 2 Hz Note that this resolution is the reciprocal of the data length NT 500 Hz is represented by k 251 5002 1 since the rst point k 1 corresponds to zero frequency 500 Hz is the same as 7500 Hz and is represented by k 3751 75002 1 Note The fast Fourier transform or t is just an ef cient algorithm for implementing the DFT The t takes advantage of symmetries in the equations describing the DFT and the periodicity of the factors eXpOZTEnkN The Inverse Discrete Fourier Transform IDFT We can reconstruct the original windowed sequence Xwn using the inverse discrete Fourier transform N l IDFTXWk ZXW keJ2 kN x39W n forn0l2Nl k0 Note that because of the periodicity of eiznknm X Wn is periodic in n with period 27 Hence this reconstruction is only equal to the original sequence Xwn over the interval 0 g n N l The two sequences are not equal outside that interval The Energy Spectrum If XIl is known to be zero outside the interval 0 g n N l that is it is nite duration we can talk about the energy of the signal Following the convention from continuous signals we commonly define total energy as the sum of the magnitudesquared of the terms in the sequence Nl 2 Total Energy 2 Xlnl l n0 Notes on the discrete Fourier transform JR Boston January 2006 page 4 A version of Parseval s theorem can be derived for the DFT namely Nl Nl 2 1 2 Zl nll Zlekl 110 N k0 following the same procedure that is used for the continuous case Parseval s theorem says that the energy in the sequence can be expressed as the sum of the squared values of the sequence itself or as the sum of the squared values of the DFT coef cients divided by N Summing the squared DFT coef cients over a speci c frequency range gives the energy over that frequency range and we sometimes refer to the coef cients lXWkl2 as the energy spectral density of the signal This is analogous to the probability density function where total probability over a range is the integral of the probability density function over that range The Periodogram In many applications we consider the sequence xn to be a random or noisy signal Examples include speech the EEG brain waves and the problem of detecting a known signal in background noise The mathematical model we use for noise is called a stochastic process and an implication of this model is that the signal has in nite duration It may seem reasonable that we could estimate the frequency content of a noisy signal by using the DTFT However because our signal model has infinite energy the DTFT does not exist Because of the random component the signal is not periodic and the discrete Fourier series does not exist What can we do Our rst thought might be to consider a nite duration of the signal and use the DFT which we know exists for any nite duration signal We could then describe the energy of this sample of the signal as a function of frequency using the energy spectrum described above The question is 7 how does this description of energy in the sample relate to the distribution of energy in the actual signal Remember that the energy in the sample is nite while the energy in the actual signal is in nite The answer to this question involves some statistics which we will only brie y summarize here You can refer to the Notes on Spectral Estimation for more detail In essence we can show that the energy spectral density of a nite sample provides an estimate of a power spectral density PSD of the actual signal Using statistical concepts we introduce a function called the autocorrelation sequence of the original random signal This sequence has the same energy content over frequency as the original signal and it also is usually nite energy meaning that the DTFT does exist Hence we can de ne an energy spectrum for the autocorrelation sequence and use this spectrum to describe the energy content over frequency of the signal Phase information is lost in the autocorrelation sequence and we consider only the magnitude spectrum Since the PSD involves random signals and is based on statistical concepts we need to think in terms of statistical estimation rather than direct calculation One method of estimating the PSD is Notes on the discrete Fourier transform IR Boston January 2006 page 5 based on the DFT of a nite duration sample of the signal For a windowed version of the original sequence Xn we calculate DFTxWn 1Exam 6W Xwk 110 and then look at the sequence 1 2 Elxwkl This sequence is called the periodogram INk It can be shown that the periodogram is equivalent to a common approach to estimating the power spectral density from the autocorrelation sequence Hence the dif culties caused by the lack of convergence of the DTFT for the original sequence XIl are avoided Technically we are using an energy spectral density to estimate a power spectral density This confusing mixture of terms is a result of the fact that the actual signal has in nite energy but nite power while our nite duration estimator has nite energy You are likely to encounter the term power spectral density or power spectrum more frequently but there are situations for which the term energy spectrum is more appropriate A basic problem in spectral estimation is a tradeoff between systematic estimation error called bias and variability of the estimate from sample to sample called variance The periodogram provides a simple intuitive method to estimate the PSD of a signal but it does not necessarily provide the best compromise between bias and variance Other estimation procedures which are mostly discussed in graduate courses are better in various ways In any case the periodogram is widely used and is a good place to start in analyzing a signal In summary the equations for the DFT and inverse DFT are Nil 2 nkN DFTXW n Z XW n e39J XW k for k 01 2 N 1 Direct Transform 110 N71 2 11km IDFTltXW k ZXW k eJ XW n for n 01 2 N l Inverse Transform k0

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