Slides2 - Sampling rate change operations: upsampling and...

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Unformatted text preview: Sampling rate change operations: upsampling and downsampling; fractional sampling; interpolation Downsampling Definition: M x[0] x[1] (↓2) x[2] x[3] x[4] M As a matrix operation: L1 L0 L0 0 0 0 M 0 1 0 M 0 0 0 0L 0L 1L M x[0] x[2] x[4] M & = M x[0] x[1] x[2] x[3] x[4] M = : x[0] x[2] x[4] : : 2 Upsampling Definition: (↑2) M x[0] x[1] x[2] M = M x[0] 0 x[1] 0 x[2] 0 M 3 As a matrix operation: L1 L0 L0 L0 L0 L0 M 0 0 1 0 0 0 M 0 0 0 0 1 0 L L L L L L M x[0] x[1] x[2] : : : = M x[0] 0 x[1] 0 x[2] 0 M 4 x[n] Downsampling Downsampling by 2 x[n] → L y[n] → ↓2 012 n y[n] L n L y[n] = x[2n] Y(ω) = ∑ L 34n 0 x[2n]e-iωn 12 = ∑ x[m]e-iωm/2 m even = ½ ∑ {1 + (-1)m} x[m]e-iωm/2 m = ½ {∑ m x[m]e-iωm 2 +∑ m ω –i( 2 + π )m} i( x[m]e ; (-1)m = e-iπm = ½ {X (ω/2) + X (ω/2 + π )} 5 Downsampling by M x[n] y[n] → ↓M → y[n] = x[Mn] Y(ω) = ∑ x[m]e-iωm/M = x[m]e-iωm/M ; 1 M M-1 –i2πm)k ∑ (e M k=0 123 123 m = nM nM M-1 1 2π = M ∑ { ∑ e–i M km } m k=0 1 if m = nM = 0 if m ≠ nM M-1 1 ω + 2 πk) M ∑ X( M k=0 6 Upsampling Upsampling by 2 x[n] → ↑2 123 123 y[n] = L y[n] → x[n/2] ; n even 0 x[n] L ; n odd 012 1 L 0 12 n y[n] L 34n Y(ω) = ∑ x[n/2]e-iωn n even = ∑ x[m]e-iω2m m = X(2ω) 7 Upsampling by L 123 123 y[n] = x[n/L] ; n = mL 0 x[n] → ↑L y[n] → ; n ≠ mL Y(ω) = ∑ x[n/L] e-iωn n=mL = X(Lω) 8 Downsampling X(ω) x[n] L L -2 -1 0 1 2 3 L L n -ω0 0 ω0 - 2π 2π ½ Xs(ω) xs[n] L nL L -2 -1 0 1 2 3 x[0] x[2] x[-2] L L L - 2π -π π 0 Y(ω) y[n] -1 0 1 y[n] = (↓2) x[n] = x[2n] π -π - 2π ↑ -2ω0 ω 2π ½ nL ω 0 Y(ω) = ½ { X( 2π ↑ 2ω0 ω)+ 2 L X( ω 2 + π)} ω 9 Upsampling 1 X(ω) x[n] L L L x[0] x[1] x[-1] - 2π n -1 0 1 L L -π -ω0 0 ω0 π L -3 -2 -1 0 1 2 3 L L n 123 123 x[n/2] ; n even y[n] = 0 ; n odd ω Y(ω) 1 y[n] 2π - 2π -π -ω0 2 ω0 2 π 2π ω Y(ω) = X(2ω) 10 10 Interpolation Use lowpass filter after upsampling x[n] u[n] ↑L H(ω) y[n] X(ω) - 2π ω0 2π 0 ω U(ω) - 2π - 2π 0 ω0 2π L 0 ω0 L ω Y(ω) 2π ω 11 11 Fractional Sampling Consider x[n] u[n] ↑L Y(ω) = = y[n] ↓M M-1 ω + 2 πk 1 ∑ U( M ) M k=0 1 M-1 2 ∑ X ( ω +M πk L) M k=0 What about x[n] ↓M d[n] ↑L y[n] Y(ω) = D(ωL) = 1 M-1 M∑X k=0 + ( ωL M 2πk ) 12 12 Basic filters, upsampling and downsampling. 14 14 15 15 16 16 17 17 18 18 19 19 20 20 ...
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