Chapter 2: The Fourier Transform

Transcription

1 EEE, EEE Part A : Digital Signal Processing Chapter Chapter : he Fourier ransform he Fourier ransform. Introduction he sampled Fourier transform of a periodic, discrete-time signal is nown as the discrete Fourier transform (DF). he DF of such a signal allows an interpretation of the frequency domain descriptions of the given signal.. Derivation of the DF he Fourier transform pair of a continuous-time signal is given by Equation. and Equation.. j πft x( t) e dt ( f ) (.) j πft x( t) ( f ) e df (.) ow consider the signal x(t) to be periodic i.e. repeating. he Fourier transform pair from above can now be represented by Equation. and Equation.4 respectively. ( f ) x( t) e jπft dt (.) j f t x ( t ) ( f ) π e (.4) where f. ow suppose that the signal x(t) is sampled times per period, as illustrated in Figure., with a sampling period of seconds. he discrete signal can now be represented by Equation., where δ is the dirac delta impulse function and has a unit area of one. x[ n ] x( t) δ ( t n ) (.) n ow replace x(t) in the Fourier integral of Equation., with x[n] of Equation.. he Equation can now be rewritten as: j πft ( f ) x( t) δ ( t n ) e dt (.6) n Since the dirac delta function δ ( t n ) Equation.7 below: for t n, Equation.6 can be modified even further to become otherwise [ f ] n x[ n ] e j πfn (.7) University of ewcastle upon yne Page.

2 EEE, EEE Part A : Digital Signal Processing Chapter : he Fourier ransform Sampled sequence δ(t-n) n ime (t) Figure.: Sampled sequence for the DF. Since f and, Equation. is formed by combining these two together. f (.) By inserting Equation. into Equation.7, the DF and its inverse, for a periodic signal, is represented by Equation.9 and Equation. respectively. [ ] n x[ n] e jπn (.9) x[ n] n [ ] e jπn (.). Digital Frequency he spectral values correspond to frequencies of f n f s and hence to digital frequencies (i.e. normalised to the sampling frequency) of: f s f s F Hence for an point transform: f F f s f s f s 4 f s 4 f s 6 f s 6 7 f s 7 University of ewcastle upon yne Page.

3 EEE, EEE Part A : Digital Signal Processing.4 Matrix Interpretation of the DF Chapter : he Fourier ransform hen woring with the DF, it is quite common to mae a substitution for the exponential term in Equation.9, such that, n e e jπn jπ n (.) Equation.9 can now be re-written and the DF can be presented in a more user-friendly fashion, as illustrated by Equation.. n n [ ] x[ n] (.) n he term is more commonly nown as a root of unity and can be represented by an argand or phasor diagram. As an example, Figure. illustrates the argand diagram for the case of an point DF Figure.: Argand diagram illustrating the th roots of unity. As the DF is a linear operation, Equation. can therefore be represented by the matrix notation defined by Equation. below. Such that the output can be derived by multiplying the corresponding th root of unity, from the phasor diagram, with the sampled signal x[n]. [] [] [] [ ] ( )... x[] ( )... x[] 4 ( )... x[]... ( ) ( )... x[ ] (.) Each of the terms in the square matrix are unit vectors with a particular angle. hese are shown graphically below, and it can be seen that each row consists of the samples of complex sine waves of different frequencies. he DF output can therefore be interpreted as the result of correlating the input samples with (complex) sine waves with frequencies equal to multiples of the fundamental (i.e. the frequency of the second row). University of ewcastle upon yne Page.

4 EEE, EEE Part A : Digital Signal Processing [] [] [] [] [] 4 [] [] 6 [] 7 x x x x x x x x [] [] [] [] [] 4 [] [] [] 6 7 Chapter : he Fourier ransform. he Fast Fourier ransform he fast Fourier transform (FF) was invented by Cooley and uey in 96. hey discovered that the DF operation could be decomposed into a number of other DFs of shorter lengths. hey then showed that the total number of computations needed for the shorter DFs was smaller than the number needed for the direct computation. In fact, the number of arithmetic operations (multiplications and additions) for the direct computation of the DF is apaproximately equal to, but for the FF algorithm reduced to approximately.log. o tae an example, if 4, the DF would require approximately 6 multiplications and additions, whilst the FF would require <, more than times fewer... Derivation of the FF he decomposition of the DF is achieved by breaing a signal x[n] down into two shorter, interleaved subsequences. his process is more commonly nown as decimation in-time (DI). Suppose a signal exists with sample values, where is an integer power of. he signal x[n] is first separated into two subsequences with / samples. One subsequence contains the samples with even-numbered values of n in x[n], and the other contains those with oddnumbered values of n. riting n(even) m and n(odd) m+, the DF from Equation. can be modified to: m m [ ] x[m] + x[m + ] (.4) m (m+ ) From the Argand diagram in Figure., it can also be shown that / i.e. now be re-written to show that it can be expressed in terms of two /-point DFs. 4 etc. Hence, the DF can m x[m] m m x[m + ] m [ ] + (.) ] [ ] + [ ] (.6) [ [] is the transform of the even numbered points in x[n], and [] is the transform of the odd-numbered points in x[n]. It is important to note that we must multiply [] by the additional term before adding it to []. his is because the sub-sequences into which we have decomposed x[n] are displaced from one another in time by one sampling interval. his term is often nown as the twiddle factor since it is a complex number of magnitude but non-zero phase, and hence merely rotates the phase of []. he computation in equation (.6) is generally broen down into so called butterfly operations. o illustrate this, suppose, and we consider by way of example the case for and + : [] [] + [] [ ] [] + [] [] [] (.6a) where is the DF of x, x, x4, x6 and is the DF of x, x, x, x7. he pair of equations in (.6a) represent the butterfly operation whose signal flow diagram is: University of ewcastle upon yne Page.4

5 EEE, EEE Part A : Digital Signal Processing Chapter : he Fourier ransform.. Radix- FF If the length transform is an integer power of, then the transform can be split into two shorter / subsequences. his process can continue until, in the limit, the transform is represented by a series of -point subsequences, each of which requires a very simple -point DF. A complete decomposition of this type gives rise to the commonly used time decimated radix- FF algorithm. A decimation-in-time FF algorithm divides up the input data into shorter interleaved subsequences. his type of FF can be performed using many butterfly operations, as illustrated in Figure. for the case of. Here it can be seen that the operations are divided up into log sections (i.e. for ). [] [] - [] [] x[] [] x[4] - [] x[] - [] x[6] - - [] x[] - [4] x[] - - [] x[] - - [6] x[7] - - ( j) j ( + j) 4 Figure.: ime-decimated radix- FF,. - [7] Decimation-in-frequency FFs are in a sense the exact opposite of the decimation-in-time algorithms; they are simply the consequence of the symmetry of the Fourier transform. A decimation-in-frequency FF, illustrated in Figure.4 for the case of, uses the opposite approach to the DI. Here, the output sequence is decimated rather than the input sequence... In-Place Computation Once the output variables for each section have been calculated, there is no longer any need for the input variables. herefore, the entire algorithm can be performed using in-place computation. hen in-place computation is used, the output sequence overwrites the input sequence in memory, for each section of the computation. hen this is done, in order for the outputs to be in the correct order, it is necessary arrange the inputs to be in bit reversed order. his can be achieved by expressing n in x[n] in binary form, reversing the order of the bits, and using the new binary number as the position in which to store that particular sample. Hence for x[6] we have n6, so that the position of x[6] will therefore be (4 th down from the top in figure. because the index starts from ). One of the problems of in place computation is that the order of addressing the data is different for each section of the FF. here are other methods of organising the storing of data such as the constant geometry method that require more memory but which have the same addressing structure for each section. University of ewcastle upon yne Page.

6 EEE, EEE Part A : Digital Signal Processing x[] Chapter : he Fourier ransform [] x[] - [4] x[] - [] x[] - - [6] x[4] - [] x[] - - [] x[6] - - [] x[7] - ( j) j ( + j) - 4 Figure.4: Frequency-decimated radix- FF,. - [7].6 indowing hen the DF is applied to an aperiodic signal it is practical to just tae a window of the sequence. A window region can be defined by effectively multiplying the signal x(n) by a rectangular window w(n), as shown in Figure. below. he windowed function of the signal x (n) can be mathematically defined by Equation.7 below. x ( n) x( n) w( n) (.7) he rectangular window function is defined by the following parameters: w( n) n n n otherwise. Random signal x(n). indowed region of x(n) n n w(n) -. n (a) -. n (b) Figure.: (a) Random signal; (b) a rectangular windowed region of x(n). University of ewcastle upon yne Page.6

7 EEE, EEE Part A : Digital Signal Processing Chapter : he Fourier ransform In DSP theory, multiplication in the time domain of two signals is equivalent to convolution in the frequency domain. Hence, Equation.7 can also be expressed by Equation., where (ω) is the frequency spectrum of the window function and (ω) is the Fourier transform of the signal. ( w) ( w) * ( w) (.) he time and frequency domain representation of a rectangular window function is illustrated in Figure.6 (a) and (b) respectively. ith reference to Figure (b), the Fourier transform of a rectangular window is the well nown sinc function. Rectangular window Sinc function A Aτ -π/τ π/τ -τ/ τ/ ime domain (a) Frequency domain (b) Figure.6: A rectangular window function; (a) ime domain. (b) Frequency domain..7 Spectral Leaage Spectral leaage is generally present when dealing with practical signals, and may lead to problems of interpretation. hen the only frequency components present are an integer multiple of the first harmonic of the DF, then all of the leaage components fall at the nulls of the sinc function. However, when at least one of the frequency components falls midway between two bins, then spectral leaage occurs. It results in a smaller pea response, plus a whole series of undesirable side lobe responses corresponding to the sidelobe peas in the spectrum of the rectangular window. o reduce the spectral leaage it is common practice to use a different window function from the rectangular window one that has a more suitable spectrum with lower side lobes. he ideal window function for (w) is of course a delta function, since the convolution operation will then not distort (w) at all. However, the inverse Fourier transform of a delta function is w(n), which is of infinite duration Choosing a suitable window always involves some ind of a trade-off between the width of the main lobe and the level of the side lobes. In practice it is desirable to have a narrow main lobe and a low side lobe level. It is also important to realise that the two cannot be achieved simultaneously and a practical trade-off between the two must be tolerated. In addition to the rectangular window there are also other window functions, such as those in the table below. hese window functions are covered in more detail in Chapter 4, and their characteristic shapes are illustrated in Figure 4.. ypical examples of window functions and their specification are as follows: ame of window function w[n] idth of main lobe (bin) Side-lobe level in (db) Rectangular.9 - Bartlett. -7 Hanning. - Hamming. -4 Blacman. -7 Kaiser β4 β β University of ewcastle upon yne Page.7

8 EEE, EEE Part A : Digital Signal Processing Chapter : he Fourier ransform ame of window function w(n) Mathematical definition Rectangular Hanning πn..cos Hamming πn.4.46 cos Blacman πn πn.4.cos +.cos Kaiser n I + β here, ( ) I x ( β ) I o x! Example: Calculate the spectral leaage of x(t), with a rectangular window truncated to samples at a sampling frequency f sam.6 Hz. Given that: x( t) sin 7. πt +. sin πt ( ) ( ) he frequency normalised to the bin width for each component is given by f f sam 7. Hence, for the frequency component f, b 6..6 And for the frequency component f, b 66.6 he frequency spectrum of a rectangular window (w) is given by the sinc function. sin( π ) ( w) π here the normalised bin width (b b ). Hence the contribution of spectral leaage for the 7. Hz signal into bin 66 is given by: sin(66 6.) π.7 (66 6.) π he frequency spectrum of the resulting leaage is illustrated in Figure.7 below. University of ewcastle upon yne Page.

9 EEE, EEE Part A : Digital Signal Processing Chapter : he Fourier ransform Spectrum of Rectangular indow - - Maximum sidelobe level idth of main lobe db Bin umber. Spectral leaage Bin number (b) Figure.7: Spectral leaage of the 7. Hz signal into bin 66. University of ewcastle upon yne Page.9

10 EEE, EEE Part A : Digital Signal Processing Chapter : he Fourier ransform. Frequency Domain Interpolation Using Zero Padding If we want to interpolate the DF output between the frequencies corresponding to the bins, then we need to evaluate the DF for more values of Ω. One way of achieving this, but still using the standard DF or FF is to add zeros to the end of a data sequence x[n], and just apply the DF. If we append (M-) zeros to end of data sequence x[n] to get xa[n], and compute the M-length DF, then this results in an output that is equivalent to computing the frequency content at M equally spaced frequency points where M>. his is shown below. he normal DF is: n n [ ] x[ n] (.9) and the zero padded DF is: M n n M [ ] x [ n] x [ n] (.) a a n a n M where the second summation has been derived by realising that x a [ n] for n M. In equation (.), we are now evaluating the DF at a frequency spacing of f s M rather than at f s for the original sequence. his process can be useful if we are trying to determine a signal component that lies between two bins, and is only a little larger than the underlying noise or spectral leaage floor illustrated below with the example in section.7. Magnitude Frequency (Hz) Magnitude Frequency (Hz) (a) Rectangular window, M (b) Rectangular window,, M496 Magnitude Frequency (Hz) Magnitude Frequency (Hz) (a) Hamming window, M (b) Hamming window,, M496 Figure.: Simulin output for example in section.7. University of ewcastle upon yne Page.