3: Discrete Cosine Transform • DFT Problems • DCT + • Basis Functions • DCT of sine wave where M is an N × 2N matrix with mk,n = cos 2π(2n+1+N)(2k+1
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3: Discrete Cosine Transform • DFT Problems • DCT + • Basis Functions • DCT of sine wave where M is an N × 2N matrix with mk,n = cos 2π(2n+1+N)(2k+1
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3: Discrete Cosine Transform
3: Discrete Cosine Transform•DFT Problems
•DCT + •Basis Functions •DCT of sine wave •DCT Properties •Energy Conservation •Energy Compaction •Frame-based coding •Lapped Transform + •MDCT (Modified DCT) •MDCT Basis Elements •Summary •MATLAB routines DSP and Digital Filters (2017-10120)Transforms: 3 - 1 / 14DFT Problems
3: Discrete Cosine Transform•DFT Problems•DCT +
•Basis Functions •DCT of sine wave •DCT Properties •Energy Conservation •Energy Compaction •Frame-based coding •Lapped Transform + •MDCT (Modified DCT) •MDCT Basis Elements •Summary •MATLAB routinesDSP and Digital Filters (2017-10120)Transforms: 3 - 2 / 14For processing 1-D or 2-D signals (especially coding), a common method is
to divide the signal into "frames" and then apply an invertible transform to each frame that compresses the information into few coefficients. The DFT has some problems when used for this purpose:DFT Problems
3: Discrete Cosine Transform•DFT Problems•DCT +
•Basis Functions •DCT of sine wave •DCT Properties •Energy Conservation •Energy Compaction •Frame-based coding •Lapped Transform + •MDCT (Modified DCT) •MDCT Basis Elements •Summary •MATLAB routinesDSP and Digital Filters (2017-10120)Transforms: 3 - 2 / 14For processing 1-D or 2-D signals (especially coding), a common method is
to divide the signal into "frames" and then apply an invertible transform to each frame that compresses the information into few coefficients. The DFT has some problems when used for this purpose:Nrealx[n]↔NcomplexX[k]: 2 real,N
2-1conjugate pairs
DFT Problems
3: Discrete Cosine Transform•DFT Problems•DCT +
•Basis Functions •DCT of sine wave •DCT Properties •Energy Conservation •Energy Compaction •Frame-based coding •Lapped Transform + •MDCT (Modified DCT) •MDCT Basis Elements •Summary •MATLAB routinesDSP and Digital Filters (2017-10120)Transforms: 3 - 2 / 14For processing 1-D or 2-D signals (especially coding), a common method is
to divide the signal into "frames" and then apply an invertible transform to each frame that compresses the information into few coefficients. The DFT has some problems when used for this purpose:Nrealx[n]↔NcomplexX[k]: 2 real,N
2-1conjugate pairs
DFT?the DTFT of a periodic signal formed by replicatingx[n].DFT Problems
3: Discrete Cosine Transform•DFT Problems•DCT +
•Basis Functions •DCT of sine wave •DCT Properties •Energy Conservation •Energy Compaction •Frame-based coding •Lapped Transform + •MDCT (Modified DCT) •MDCT Basis Elements •Summary •MATLAB routinesDSP and Digital Filters (2017-10120)Transforms: 3 - 2 / 14For processing 1-D or 2-D signals (especially coding), a common method is
to divide the signal into "frames" and then apply an invertible transform to each frame that compresses the information into few coefficients. The DFT has some problems when used for this purpose:Nrealx[n]↔NcomplexX[k]: 2 real,N
2-1conjugate pairs
DFT?the DTFT of a periodic signal formed by replicatingx[n]. ?Spurious frequency components from boundary discontinuity. N=20 f=0.08DFT Problems
3: Discrete Cosine Transform•DFT Problems•DCT +
•Basis Functions •DCT of sine wave •DCT Properties •Energy Conservation •Energy Compaction •Frame-based coding •Lapped Transform + •MDCT (Modified DCT) •MDCT Basis Elements •Summary •MATLAB routinesDSP and Digital Filters (2017-10120)Transforms: 3 - 2 / 14For processing 1-D or 2-D signals (especially coding), a common method is
to divide the signal into "frames" and then apply an invertible transform to each frame that compresses the information into few coefficients. The DFT has some problems when used for this purpose:Nrealx[n]↔NcomplexX[k]: 2 real,N
2-1conjugate pairs
DFT?the DTFT of a periodic signal formed by replicatingx[n]. ?Spurious frequency components from boundary discontinuity. N=20 f=0.08 The Discrete Cosine Transform (DCT) overcomes these problems. DCT+3: Discrete Cosine Transform
•DFT Problems•DCT +•Basis Functions •DCT of sine wave •DCT Properties •Energy Conservation •Energy Compaction •Frame-based coding •Lapped Transform + •MDCT (Modified DCT) •MDCT Basis Elements •Summary •MATLAB routinesDSP and Digital Filters (2017-10120)Transforms: 3 - 3 / 14To form the Discrete Cosine Transform (DCT), replicatex[0 :N-1]but in
reverse order and insert a zero between each pair of samples:0 12 23y[r]
Take the DFT of length4Nreal, symmetric, odd-sample-only sequence. DCT+3: Discrete Cosine Transform
•DFT Problems•DCT +•Basis Functions •DCT of sine wave •DCT Properties •Energy Conservation •Energy Compaction •Frame-based coding •Lapped Transform + •MDCT (Modified DCT) •MDCT Basis Elements •Summary •MATLAB routinesDSP and Digital Filters (2017-10120)Transforms: 3 - 3 / 14To form the Discrete Cosine Transform (DCT), replicatex[0 :N-1]but in
reverse order and insert a zero between each pair of samples:0 12 23y[r]
Take the DFT of length4Nreal, symmetric, odd-sample-only sequence.Result is real, symmetric and anti-periodic:
01223Y[k]
DCT+3: Discrete Cosine Transform
•DFT Problems•DCT +•Basis Functions •DCT of sine wave •DCT Properties •Energy Conservation •Energy Compaction •Frame-based coding •Lapped Transform + •MDCT (Modified DCT) •MDCT Basis Elements •Summary •MATLAB routinesDSP and Digital Filters (2017-10120)Transforms: 3 - 3 / 14To form the Discrete Cosine Transform (DCT), replicatex[0 :N-1]but in
reverse order and insert a zero between each pair of samples:0 12 23y[r]
Take the DFT of length4Nreal, symmetric, odd-sample-only sequence. Result is real, symmetric and anti-periodic: only need firstNvalues 01223Y[k]
÷2-→
DCT+3: Discrete Cosine Transform
•DFT Problems•DCT +•Basis Functions •DCT of sine wave •DCT Properties •Energy Conservation •Energy Compaction •Frame-based coding •Lapped Transform + •MDCT (Modified DCT) •MDCT Basis Elements •Summary •MATLAB routinesDSP and Digital Filters (2017-10120)Transforms: 3 - 3 / 14To form the Discrete Cosine Transform (DCT), replicatex[0 :N-1]but in
reverse order and insert a zero between each pair of samples:0 12 23y[r]
Take the DFT of length4Nreal, symmetric, odd-sample-only sequence. Result is real, symmetric and anti-periodic: only need firstNvalues 01223Y[k]
÷2-→
Forward DCT:
XC[k] =?N-1
n=0x[n]cos2π(2n+1)k4Nfork= 0 :N-1
DCT+3: Discrete Cosine Transform
•DFT Problems•DCT +•Basis Functions •DCT of sine wave •DCT Properties •Energy Conservation •Energy Compaction •Frame-based coding •Lapped Transform + •MDCT (Modified DCT) •MDCT Basis Elements •Summary •MATLAB routinesDSP and Digital Filters (2017-10120)Transforms: 3 - 3 / 14To form the Discrete Cosine Transform (DCT), replicatex[0 :N-1]but in
reverse order and insert a zero between each pair of samples: