3: Discrete Cosine Transform • DFT Problems • DCT + • Basis Functions • DCT of sine wave • DCT Properties • Energy Conservation • Energy Compaction • Frame-based coding • LappedTransform + • MDCT (Modified DCT) • MDCT Basis Elements • Summary 3: Discrete Cosine Transform • MATLAB routines DSP and Digital Filters (2017-10120) Transforms: 3 – 1 / 14 DFT Problems 3: Discrete Cosine Transform For processing 1-D or 2-D signals (especially coding), a common method is • DFT Problems • DCT + to divide the signal into “frames” and then apply an invertible transform to • Basis Functions • DCT of sine wave each frame that compresses the information into few coefficients. • DCT Properties • Energy Conservation The DFT has some problems when used for this purpose: • Energy Compaction • Frame-based coding • LappedTransform + • MDCT (Modified DCT) • MDCT Basis Elements • Summary • MATLAB routines DSP and Digital Filters (2017-10120) Transforms: 3 – 2 / 14 DFT Problems 3: Discrete Cosine Transform For processing 1-D or 2-D signals (especially coding), a common method is • DFT Problems • DCT + to divide the signal into “frames” and then apply an invertible transform to • Basis Functions • DCT of sine wave each frame that compresses the information into few coefficients. • DCT Properties • Energy Conservation The DFT has some problems when used for this purpose: • Energy Compaction • Frame-based coding N real x n N complex X k : 2 real, N conjugate pairs • LappedTransform + • [ ] ↔ [ ] 2 − 1 • MDCT (Modified DCT) • MDCT Basis Elements • Summary → • MATLAB routines DSP and Digital Filters (2017-10120) Transforms: 3 – 2 / 14 DFT Problems 3: Discrete Cosine Transform For processing 1-D or 2-D signals (especially coding), a common method is • DFT Problems • DCT + to divide the signal into “frames” and then apply an invertible transform to • Basis Functions • DCT of sine wave each frame that compresses the information into few coefficients. • DCT Properties • Energy Conservation The DFT has some problems when used for this purpose: • Energy Compaction • Frame-based coding N real x n N complex X k : 2 real, N conjugate pairs • LappedTransform + • [ ] ↔ [ ] 2 − 1 • MDCT (Modified DCT) • MDCT Basis Elements • Summary → • MATLAB routines • DFT ∝ the DTFT of a periodic signal formed by replicating x[n] . DSP and Digital Filters (2017-10120) Transforms: 3 – 2 / 14 DFT Problems 3: Discrete Cosine Transform For processing 1-D or 2-D signals (especially coding), a common method is • DFT Problems • DCT + to divide the signal into “frames” and then apply an invertible transform to • Basis Functions • DCT of sine wave each frame that compresses the information into few coefficients. • DCT Properties • Energy Conservation The DFT has some problems when used for this purpose: • Energy Compaction • Frame-based coding N real x n N complex X k : 2 real, N conjugate pairs • LappedTransform + • [ ] ↔ [ ] 2 − 1 • MDCT (Modified DCT) • MDCT Basis Elements • Summary → • MATLAB routines • DFT ∝ the DTFT of a periodic signal formed by replicating x[n] . ⇒ Spurious frequency components from boundary discontinuity. → N=20 f=0.08 DSP and Digital Filters (2017-10120) Transforms: 3 – 2 / 14 DFT Problems 3: Discrete Cosine Transform For processing 1-D or 2-D signals (especially coding), a common method is • DFT Problems • DCT + to divide the signal into “frames” and then apply an invertible transform to • Basis Functions • DCT of sine wave each frame that compresses the information into few coefficients. • DCT Properties • Energy Conservation The DFT has some problems when used for this purpose: • Energy Compaction • Frame-based coding N real x n N complex X k : 2 real, N conjugate pairs • LappedTransform + • [ ] ↔ [ ] 2 − 1 • MDCT (Modified DCT) • MDCT Basis Elements • Summary → • MATLAB routines • DFT ∝ the DTFT of a periodic signal formed by replicating x[n] . ⇒ Spurious frequency components from boundary discontinuity. → N=20 f=0.08 The Discrete Cosine Transform (DCT) overcomes these problems. DSP and Digital Filters (2017-10120) Transforms: 3 – 2 / 14 DCT + 3: Discrete Cosine Transform • DFT Problems To form the Discrete Cosine Transform (DCT), replicate x[0 : N − 1] but in • DCT + reverse order and insert a zero between each pair of samples: • Basis Functions • DCT of sine wave y[r] • DCT Properties • Energy Conservation • Energy Compaction → • Frame-based coding • LappedTransform + • MDCT (Modified DCT) 0 12 23 • MDCT Basis Elements Take the DFT of length 4N real, symmetric, odd-sample-only sequence. • Summary • MATLAB routines DSP and Digital Filters (2017-10120) Transforms: 3 – 3 / 14 DCT + 3: Discrete Cosine Transform • DFT Problems To form the Discrete Cosine Transform (DCT), replicate x[0 : N − 1] but in • DCT + reverse order and insert a zero between each pair of samples: • Basis Functions • DCT of sine wave y[r] • DCT Properties • Energy Conservation • Energy Compaction → • Frame-based coding • LappedTransform + • MDCT (Modified DCT) 0 12 23 • MDCT Basis Elements Take the DFT of length 4N real, symmetric, odd-sample-only sequence. • Summary • MATLAB routines Result is real, symmetric and anti-periodic: Y[k] 12 0 23 DSP and Digital Filters (2017-10120) Transforms: 3 – 3 / 14 DCT + 3: Discrete Cosine Transform • DFT Problems To form the Discrete Cosine Transform (DCT), replicate x[0 : N − 1] but in • DCT + reverse order and insert a zero between each pair of samples: • Basis Functions • DCT of sine wave y[r] • DCT Properties • Energy Conservation • Energy Compaction → • Frame-based coding • LappedTransform + • MDCT (Modified DCT) 0 12 23 • MDCT Basis Elements Take the DFT of length 4N real, symmetric, odd-sample-only sequence. • Summary • MATLAB routines Result is real, symmetric and anti-periodic: only need first N values Y[k] 12 ÷2 0 23 −→ DSP and Digital Filters (2017-10120) Transforms: 3 – 3 / 14 DCT + 3: Discrete Cosine Transform • DFT Problems To form the Discrete Cosine Transform (DCT), replicate x[0 : N − 1] but in • DCT + reverse order and insert a zero between each pair of samples: • Basis Functions • DCT of sine wave y[r] • DCT Properties • Energy Conservation • Energy Compaction → • Frame-based coding • LappedTransform + • MDCT (Modified DCT) 0 12 23 • MDCT Basis Elements Take the DFT of length 4N real, symmetric, odd-sample-only sequence. • Summary • MATLAB routines Result is real, symmetric and anti-periodic: only need first N values Y[k] 12 ÷2 0 23 −→ N−1 2π(2n+1)k Forward DCT: XC [k] = n=0 x[n] cos 4N for k =0: N − 1 P DSP and Digital Filters (2017-10120) Transforms: 3 – 3 / 14 DCT + 3: Discrete Cosine Transform • DFT Problems To form the Discrete Cosine Transform (DCT), replicate x[0 : N − 1] but in • DCT + reverse order and insert a zero between each pair of samples: • Basis Functions • DCT of sine wave y[r] • DCT Properties • Energy Conservation • Energy Compaction → • Frame-based coding • LappedTransform + • MDCT (Modified DCT) 0 12 23 • MDCT Basis Elements Take the DFT of length 4N real, symmetric, odd-sample-only sequence. • Summary • MATLAB routines Result is real, symmetric and anti-periodic: only need first N values Y[k] 12 ÷2 0 23 −→ N−1 2π(2n+1)k Forward DCT: XC [k] = n=0 x[n] cos 4N for k =0: N − 1 1 P 2 N−1 2π(2n+1)k Inverse DCT: x[n] = N X[0] + N k=1 X[k] cos 4N P DSP and Digital Filters (2017-10120) Transforms: 3 – 3 / 14 Basis Functions 3: Discrete Cosine Transform kn 1 N−1 j2π N • DFT Problems DFT basis functions: x[n] = N k=0 X[k]e • DCT + • Basis Functions P • DCT of sine wave • DCT Properties • Energy Conservation • Energy Compaction • Frame-based coding • LappedTransform + • MDCT (Modified DCT) • MDCT Basis Elements • Summary • MATLAB routines DSP and Digital Filters (2017-10120) Transforms: 3 – 4 / 14 Basis Functions 3: Discrete Cosine Transform kn 1 N−1 j2π N • DFT Problems DFT basis functions: x[n] = N k=0 X[k]e • DCT + • Basis Functions P • DCT of sine wave • DCT Properties • Energy Conservation • Energy Compaction • Frame-based coding • LappedTransform + • MDCT (Modified DCT) • MDCT Basis Elements • Summary • MATLAB routines DSP and Digital Filters (2017-10120) Transforms: 3 – 4 / 14 Basis Functions 3: Discrete Cosine Transform kn 1 N−1 j2π N • DFT Problems DFT basis functions: x[n] = N k=0 X[k]e • DCT + • Basis Functions P • DCT of sine wave • DCT Properties • Energy Conservation • Energy Compaction • Frame-based coding • LappedTransform + • MDCT (Modified DCT) • MDCT Basis Elements • Summary • MATLAB routines DSP and Digital Filters (2017-10120) Transforms: 3 – 4 / 14 Basis Functions 3: Discrete Cosine Transform kn 1 N−1 j2π N • DFT Problems DFT basis functions: x[n] = N k=0 X[k]e • DCT + • Basis Functions P • DCT of sine wave • DCT Properties • Energy Conservation • Energy Compaction • Frame-based coding • LappedTransform + • MDCT (Modified DCT) • MDCT Basis Elements • Summary • MATLAB routines DSP and Digital Filters (2017-10120) Transforms: 3 – 4 / 14 Basis Functions 3: Discrete Cosine Transform kn 1 N−1 j2π N • DFT Problems DFT basis functions: x[n] = N k=0 X[k]e • DCT + • Basis Functions P • DCT of sine wave • DCT Properties • Energy Conservation • Energy Compaction • Frame-based coding • LappedTransform + • MDCT (Modified DCT) • MDCT Basis Elements • Summary • MATLAB routines DSP and Digital Filters (2017-10120) Transforms: 3 – 4 / 14 Basis Functions 3: Discrete Cosine Transform kn 1 N−1 j2π N • DFT Problems DFT basis functions: x[n] = N k=0 X[k]e • DCT + • Basis Functions P • DCT of sine wave • DCT Properties • Energy Conservation • Energy Compaction • Frame-based coding • LappedTransform + • MDCT (Modified DCT) • MDCT Basis Elements