Down-Sampling (4B) /5/
Copyright (c) 9,, Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version. or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover exts, and no Back-Cover exts. A copy of the license is included in the section entitled "GNU Free Documentation License". Please send corrections (or suggestions) to youngwlim@hotmail.com. his document was produced by using OpenOffice and Octave. /5/
Decreasing Sampling Frequency DOWN 4 3 f f 4 s f f f 4 s 4 s 4 s s 3 Sampling Frequency Sampling Frequency f ' s = Sampling ime = Sampling ime f ' = s 4 4B Down-Sampling 3 /5/
Coarse Sequence & Spectrum x[n] y[n] ime Sequence DOWN 4 = 4 = 4 Linear Frequency 3 f f 4 s f f f 4 s 4 s 4 s s Linear Frequency 3 ω π 3 π π π π π 3 π π Normalized Radian Frequency Normalized Radian Frequency ω 4B Down-Sampling 4 /5/
Normalized Radian Frequency x[n] y[n] ime Sequence f B ω = ω s = ω / s ω = ω = π f 4 f = f B / 4 = f B 4 he Same ime Sequence Normalized Radian Frequency f = 4 f B / = f B 4 he Highest Frequency: y ' [n] ime Sequence 4 f B Normalized to Normalized Radian Frequency 4B Down-Sampling 5 /5/
Coarse Sequence Spectrum Linear Frequency x[n] = DOWN f B High Freq 3 y[n] ime Sequence 4 = 4 4 3 = he Same ime Sequence f B High Freq y[n] ime Sequence = f B 3 High Freq 4B Down-Sampling 6 /5/
Coarse Sequence Spectrum Normalized Frequency x[n] Normalized Radian Frequency DOWN f B π 3 π π π ω y[n] ime Sequence Normalized Radian Frequency 4 π 3 π π π ω = he Same ime Sequence f B 4 y[n] ime Sequence Normalized Radian Frequency π 3 π π 4 f B π ω 4B Down-Sampling 7 /5/
Coarse Sequence Spectrum x[n] = Normalized Radian Frequency DOWN f B High Freq 3 f B π 3 π π π ω y [n] ime Sequence 4 = 4 Normalized Radian Frequency = 4 he Same ime Sequence f B High Freq 3 π 3 π π f B 4 π ω y[ n] ime Sequence = Normalized Radian Frequency f B 3 High Freq π π 3 π 4 f B π ω 4B Down-Sampling 8 /5/
Coarse Sequence Generation x[n] x d [n] 4 Sampling Period x[n] x d [n] Remove Zero Samples y[n] p[n] 4 Sampling Period p[n] y[n] ime Sequence t 4 4B Down-Sampling 9 /5/
Down Sampling in wo Steps x[n] v[n] 4 Sampling Period x[n] Suppress D Samples v[n] Remove D Zeros & Compact y[n] δ D [n] 4 f B Highest Frequency Sampling Period δ D [n] D y[n] ime Sequence D t 4B Down-Sampling /5/
Down-Sampling Operator x[n] δ D [n] D v[n] D D t 4 D x[n] x[n] D y[n] y[n] D-fold Decimator y[n] = S /D x[n] = x[n D] y[] = x[ D] y[] = x[ D] y[3] = x[3 D] 4B Down-Sampling /5/
Suppressing D- Samples x[n] δ D [n] D v[n] D D t 4 D δ D [n] = if mod(n, D) = otherwise v[n] = δ D [n] x[n] V [ z] = + v[] z + v[ D] z D + v[d] z D + V [ z] = + n= + v[n] z n = m= v[m D] z m D = F (z D ) 4B Down-Sampling /5/
Example When D= () x[n] δ [n] D = v[n] t 4 n x[n] = e j ωn δ [n] = ( + ( )n ) = ( + e j π n ) (e j π = ) v [n] = x[ n] + e jπ n x[n] = e j ω n + e j π n e j ωn = e j ω n + e+ j(ω π) n V ( z) = + n= ( x[n] z n + x[n]( z) n ) = X (z ) + X ( z) V (e j ω ) = X (e j ω ) + X (e j π e j ω ) V ( ω) = X ( ω) + X ( ω π) 4B Down-Sampling 3 /5/
Example When D= () x[n] X ( ω) π( / ) π 3 π π π x[n] X ( ω π) π( / ) π 3 π π π v[n] V ( ω) π( / ) 4 π 3 π π π v [n] = x[ n] + e jπ n x[n] V ( z) = X ( z) + X ( z) V (e j ω ) = X (e j ω ) + X (e j π e j ω ) V ( ω) = X ( ω) + X ( ω π) 4B Down-Sampling 4 /5/
Example When D= (3) x[n] v[n] 4 v[n] 4 4B Down-Sampling 5 /5/
Z-ransform & Suppressing D- Samples () x[n] δ D [n] D v[n] D D t 4 D D δ D [n] = D k = e j π k n/ D = D D D = if mod(n, D) = e j π n = j π n/ D e otherwise v[n] = δ D [n] x[n] D v[n] = δ D [n] x[n] = D k= j π k n/ D x[n]e Z {x[n]e j ω k n } = + x[n](e j ω k z) n n= = X (e jω k z) D V ( z) = D k= X (e j π k/ D z) ω k = π k / D e j π k / D z = e j π k /D e j ω = e j( ω π k/ D) D V ( ω) = D k= X ( ω π k / D) 4B Down-Sampling 6 /5/
Z-ransform & Suppressing D- Samples () x[n] δ D [n] D v[n] D D t 4 D D V ( z) = D k= X (e j π k/ D z) D V ( ω) = D k= X ( ω π k / D) X ( f ) V ( f ) 4 3 f f f 3 f f 4 s 4 s 4 s 4 s s X ( ω) π( / ) V ( ω) π( / ) 4 π 3 π π π π 3 π π π 4B Down-Sampling 7 /5/
Z-ransform & Removing D- Zeros () v[n] D y[n] ime Sequence f B y[n] ime Sequence 4 f B 4 D 4 Y (z) = + y[] + y[] z + y[] z + + y[n] z n + = + v[] + v[ D] z + v [ D] z + + v[n D] z n + Y (z ) = V ( z / D ) D V ( z) = D k= X (e j π k/ D z) D Y (z) = D k= X (e j π k/ D z /D ) D V ( ω) = D k= X ( ω π k / D) D V ( ω) = X ( ω/ D π k / D) D k= 4B Down-Sampling 8 /5/
Z-ransform & Removing D- Zeros () v[n] D y[n] ime Sequence f B y[n] ime Sequence 4 f B 4 D 4 D V ( z) = D k= X (e j π k/ D z) D Y (z) = D k= X (e jπ k/ D z /D ) D V ( ω) = D k= X ( ω π k / D) D V ( ω) = D k= X ( ω/ D π k / D) V ( f ) V ( f ) V ( f ) /4 4 4 4 3 3 3 V ( ω) π( / ) V ( ω) π( / ) V ( ω) π( / ) 4 4 4 π 3 π π π π 3 π π π π 3 π π π 4B Down-Sampling 9 /5/
Example When D= (3) x[n] v[n] v[n] v[n] v[n] 4B Down-Sampling /5/
Down Sampling x[n] v[n] y[n] ime Sequence 4 Sampling Period x[n] Suppress D Samples v[n] Remove D Zeros y[n] Ideal Sampling δ D [n] Amplitude Axis Scaling / D ime Axis Scaling / D Spectrum Replication / D Stretching Spectrum ω / D 4B Down-Sampling /5/
Measuring Rotation Rate 4B Down-Sampling /5/
Signals with Harmonic Frequencies () Hz cycle / sec Hz cycles / sec 3 Hz 3 cycles / sec 4 Hz 4 cycles / sec 5 Hz 5 cycles / sec 6 Hz 6 cycles / sec 7 Hz 7 cycles / sec cos ( πt) cos ( π t) cos (3 πt) cos (4 πt) cos (5 πt) cos (6 πt ) cos (7 πt) = e+ j( π)t j( π)t + e = e+ j( π)t j( π)t + e = e+ j(3 π)t + e j(3 π)t = e+ j(4 π)t j( 4 π) t + e = e+ j(5 π)t + e j(5 π)t = e+ j(6 π)t + e j(6 π)t = e+ j(7 π)t + e j(7 π)t 4B Down-Sampling 3 /5/
Signals with Harmonic Frequencies () Hz cycle / sec Hz cycles / sec 3 Hz 3 cycles / sec 4 Hz 4 cycles / sec 5 Hz 5 cycles / sec 6 Hz 6 cycles / sec 7 Hz 7 cycles / sec 4 6 8 4 4 6 8 4 4B Down-Sampling 4 /5/
Sampling Frequency Hz sample / sec Hz samples / sec 3 Hz 3 samples / sec 4 Hz 4 samples / sec 5 Hz 5 samples / sec 6 Hz 6 samples / sec 7 Hz 7 samples / sec 4 6 8 4 4 6 8 4 4B Down-Sampling 5 /5/
Nyquist Frequency Hz cycle / sec Hz cycles / sec 3 Hz 3 cycles / sec 4 Hz 4 cycles / sec 5 Hz 5 cycles / sec 6 Hz 6 cycles / sec 7 Hz 7 cycles / sec x sample / sec x samples / sec x3 samples / sec x4 samples / sec x5 samples / sec x6 samples / sec x7 samples / sec 4B Down-Sampling 6 /5/
Aliasing Hz 7 Hz 6π x4 samples / sec 4 4 π Hz 6 Hz 4 4 6π x4 samples / sec 6π 3 Hz 5 Hz 6 6 6π x4 samples / sec 6π 4 Hz 8 8 6π x4 samples / sec 4B Down-Sampling 7 /5/
Sampling s = rad /sec = f = f = s rad / sec = s rad / sec f = rad / sec f = rad / sec rad / s sec rad / s sec rad / s sec 4B Down-Sampling 8 /5/
Sampling = f rad /sec s = rad /sec s = rad / s sec rad / s sec For the period of s Angular displacement = s rad rad = f s rad = 4 s rad = rad 4B Down-Sampling 9 /5/
Angular Frequencies in Sampling continuous-time signals Signal Frequency f = Signal Angular Frequency ω = π f (rad / sec) For second For revolution ω = π f (rad / sec) π (rad ) / (sec) sampling sequence Sampling Frequency = s For second For revolution ω s = π (rad /sec) π (rad ) / s (sec) Sampling Angular Frequency s = rad /sec 4B Down-Sampling 3 /5/
4B Down-Sampling 3 /5/
4B Down-Sampling 3 /5/
4B Down-Sampling 33 /5/
References [] http://en.wikipedia.org/ [] J.H. McClellan, et al., Signal Processing First, Pearson Prentice Hall, 3 [3] A graphical interpretation of the DF and FF, by Steve Mann [4] R. Cristi, Modern Digital Signal Processing /5/