Coputation of the DFT of In ost practical applications, sequences of interest are real In such cases, the syetry properties of the DFT given in Table 5. can be exploited to ake the DFT coputations ore efficient -Point DFTs of Two Length- Letg[ and be two length- real sequences with G[ and H[ denoting their respective -point DFTs These two -point DFTs can be coputed efficiently using a single -point DFT Define a coplex length- sequence x [ g[ + j Hence, g[ Re{} and I{} -Point DFTs of Two Length- LetX[ denote the -point DFT of Then, fro Table 5. we arrive at ote that X [ k ] X *[ k 3 j G [ { X[ + X *[ k H [ { X[ X *[ k ]} * ]} ] -Point DFTs of Two Length- Exaple-We copute the 4-point DFTs of the two real sequences g[ and given below { g[ } { }, { } { } Then { x [ } { g[ } + j{ } is given by 4 { x [ } { + j + j j + j} -Point DFTs of Two Length- Its DFTX[ is X[ + j 4 + j6 X[ j j + j X[ j X[ j j + j j Fro the above X *[ [ 4 j6 j Hence X *[ 4 k [4 j6 j 5 -Point DFTs of Two Length- Therefore { G [ } { 4 j + j} { H [ } { 6 j + j} verifying the results derived earlier 6
-Point DFT of a Real Let v[ be a length- real sequence with an -point DFT V[ Define two length- real sequences g[ and as follows: g[ v[, v[ n +, n LetG[ and H[ denote their respective - point DFTs -Point DFT of a Real Define a length- coplex sequence { x [ } { g[ } + j{ } with an -point DFT X[ Then as shown earlier G [ { X[ + X *[ k j H [ { X[ X *[ k ]} ]} 7 8 -Point DFT of a Real ow 9 W nk nk ( n+ ) k W + v[ n + W nk nk k ] W + n W W nk k nk + W W k V [ v[ v[ g [ n ] g [ W, -Point DFT of a Real i.e., k V[ G[ k ] + W H[ k ], k Exaple-Let us deterine the 8-point DFT V[ of the length-8 real sequence We for two length-4 real sequences as follows { v[ } { } -Point DFT of a Real ow k V[ G[ k + W8 H[ k, k 7 Substituting the values of the 4-point DFTs G[ and H[ coputed earlier we get { g[ } { v[ } { } { } { v[ n + } { } -Point DFT of a Real V[ ] G[ + H[ 4 + 6 V [ G[ + W 8 H[ ( + e jπ / 4 ( j. 44 jπ / V[ G[ + W8 H[ + e 3 V [ G[ + W 8 H[ j3π / 4 ( + + e ( + j. 44 4 jπ V [ G[ + W8 H[ 4 + e 6
-Point DFT of a Real 5 V [ G [ + W 8 H [ j5π / 4 ( + e ( + j. 44 6 j π / V[ 6] G[ + W8 H[ + e 7 V [ 7] G [ + W 8 H [ j7π / 4 ( + + e ( + + j. 44 3 Linear Convolution Using the DFT Linear convolution is a key operation in any signal processing applications Since a DFT can be efficiently ipleented using FFT algoriths, it is of interest to develop ethods for the ipleentation of linear convolution using the DFT 3 4 5 Linear Convolution of Two Finite-Length Sequences Letg[ and be two finite-length sequences of length and M, respectively Denote L + M Define two length-l sequences g[, g e [,, h e [, n n L n M M n L 6 Linear Convolution of Two Finite-Length Sequences Then yl [ g[ * yc[ g[ L The corresponding ipleentation schee is illustrated below g[ Zero-padding g e [ ( + M ) with Length- ( M ) zeros point DFT ( + M ) y L [ Zero-padding h e [ ( + M ) point IDFT with Length-M ( ) zeros point DFT Length- ( + M ) Linear Convolution of a Finite- Length Sequence with an Infinite-Length Sequence We next consider the DFT-based ipleentation of 7 M y[ l ] n l] * l where is a finite-length sequence of length M and is an infinite length (or a finite length sequence of length uch greater than M) 8 We first segent, assued to be a causal sequence here without any loss of generality, into a set of contiguous finitelength subsequences x [ of length each: x x [ [ n ] where n + ], x [, n otherwise 3
9 Thus we can write y [ * y[ n ] where y [ * x[ Since is of length M and x [ is of length, the linear convolution * x[ is of length + M As a result, the desired linear convolution y [ * has been broken up into a su of infinite nuber of short-length linear convolutions of length + M each: y [ x[ * Each of these short convolutions can be ipleented using the DFT-based ethod discussed earlier, where now the DFTs (and the IDFT) are coputed on the basis of ( + M ) points There is one ore subtlety to take care of before we can ipleent y [ y [ n ] using the DFT-based approach ow the first convolution in the above su, y [ * x[, is of length + M and is defined for n + M The second short convolution y [ * x[, is also of length + M but is defined for n + M There is an overlap of M saples between these two short linear convolutions Likewise, the third short convolution y [ * x[, is also of length + M but is defined for n 3 + M Thus there is an overlap of M saples between * x[ and * x[ In general, there will be an overlap of M saples between the saples of the short convolutions * xr [ and * xr[ for This process is illustrated in the figure on the next slide for M 5 and 7 3 4 4
Add Add Therefore, y[ obtained by a linear convolution of and is given by y[ y[, n 6 y[ y[ + y[ n 7], 7 n y[ y[ n 7], y[ y[ n 7] + y[ n, n 3 4 n 7 y[ y [ n, 8 n 5 6 The above procedure is called the overlapadd ethod since the results of the short linear convolutions overlap and the overlapped portions are added to get the correct final result The function fftfilt can be used to ipleent the above ethod Progra 5_5 illustrates the use of fftfilt in the filtering of a noise-corrupted signal using a length-3 oving average filter The plots generated by running this progra is shown below Aplitude 8 6 4 s[ y[ 7 8-3 4 5 6 Tie index n 9 In ipleenting the overlap-add ethod using the DFT, we need to copute two ( + M ) -point DFTs and one ( + M ) - point IDFT since the overall linear convolution was expressed as a su of short-length linear convolutions of length ( + M ) each It is possible to ipleent the overall linear convolution by perforing instead circular convolution of length shorter than ( + M ) 3 To this end, it is necessary to segent into overlapping blocks x [, keep the ters of the circular convolution of with x [ that corresponds to the ters obtained by a linear convolution of and x [, and throw away the other parts of the circular convolution 5
To understand the correspondence between the linear and circular convolutions, consider a length-4 sequence and a length-3 sequence Let y L [ denote the result of a linear convolution of with The six saples of y L [ are given by y L [ y L [ y L [ y L [ y L [ y L [ 3 3 33 If we append with a single zero-valued saple and convert it into a length-4 sequence h e [, the 4-point circular convolution y C [ of h e [ and is given by y C [ y C [ y C [ y C [ 34 If we copare the expressions for the saples of y L [ with the saples of y C] [, we observe that the first ters of yc[n do not correspond to the first ters of y L [, whereas the last ters of y C [ are precisely the sae as the 3rd and 4th ters of y L [, i.e., y [ ] y [, y [ ] y [ L C L yc L C L yc y [ [, y [ [ 35 General case: -point circular convolution of a length-m sequence with a length- sequence with > M First M saples of the circular convolution are incorrect and are rejected Reaining M + saples correspond to the correct saples of the linear convolution of with 36 ow, consider an infinitely long or very long sequence Break it up as a collection of saller length (length-4) overlapping sequences x [ as x [ n + ], n 3, ext, for w [ x [ 4 6
37 w [ w [ w y[ w y[ w [ 3 w [ w y[ x w y[ 38 Or, equivalently, w [ ] x [ x [ x [ w [ ] x [ x [ x [ w [ ] x [ x [ x [ w [ ] x [ x [ x [ Coputing the above for,,, 3,..., and substituting the values of [ we arrive at [ [ [ [ w [ [ 6] x [ 7] [ 6] y[ 6] [ 7] 6] y[ 7] w + w w It should be noted that to deterine y[ and y[, we need to for x [: x [ ], x [, x [, x [ and copute w [ 4 x [ for n 3 reject w [ and w [ ], and save w [ y[ and w [ [ y 39 4 General Case: Let be a length- sequence Let x [ denote the -th section of an infinitely long sequence of length and defined by x [ n + ( + )], n with M < 4 Letw[ x[ Then, we reject the first M saples of w [ and abut the reaining M + saples of w [ to for y L [, the linear convolution of and If y [ denotes the saved portion of w [, i.e., n M y[ w[, M n 4 7
43 Then yl[ n + ( M + )] y[, M n The approach is called overlap-save ethod since the input is segented into overlapping sections and parts of the results of the circular convolutions are saved and abutted to deterine the linear convolution result 44 Process is illustrated next 45 8