Down-Sampling (B) /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/
= Spectrum Replication () Ideal Sampling x(t) x(t) = + n = x(n ) δ(t n ) X t s(t) = + δ(t n ) n = s(t) = + e + j π m t m = t x(t) = + n = x(t) e + j π m t x(t) Shift Property t X ( f ) = + m = X ( f m ) B Down-Sampling /5/
= Spectrum Replication () S ( f ) = + δ( f m ) m = Frequency Domain X ( f ) A Convolution in Frequency X ( f ) = X ( f ) S ( f ) ω + = X ( f f ' )S ( f ' ) d f ' S ( f ) * = + m = + X ( f f ' )δ( f ' m ) d f ' π π + π + π ω X ( f ) = + m = X ( f m ) X ( f ) A π π + π + π ω B Down-Sampling /5/
Decreasing Sampling Frequency F s = DOWN / / F s ' = F s ω/ π f f s f f f s s s s F s ' ω/ π Sampling Frequency Sampling ime F s = Sampling Frequency F s ' = = Sampling ime ' = B Down-Sampling 5 /5/
Coarse Sequence & Spectrum y[n] ime Sequence DOWN F s = = / / F s ' = = Linear Frequency F s ω/ π f f s f f f s s s s / F s / = expanded Linear Frequency F s ' = 8 F s ' ω/ π ω π π Normalized Radian Frequency Normalized Radian Frequency ω B Down-Sampling 6 /5/
Normalized Radian Frequency y[n] ime Sequence f B ω = ω s = ω / s ω = ω = π f f = f B / = f B he Same ime Sequence Normalized Radian Frequency f = f B / = f B he Highest Frequency: y ' [n] ime Sequence f B Normalized to Normalized Radian Frequency B Down-Sampling 7 /5/
Coarse Sequence Spectrum Linear Frequency BW / Fs f B = 8 = y[n] DOWN ime Sequence BW f B = 8 / f B High Freq Fs = = he Same ime Sequence f B High Freq y[n] ime Sequence BW / Fs f B = = f B High Freq B Down-Sampling 8 /5/
Coarse Sequence Spectrum Normalized Frequency BW / Fs f B = 8 = y[n] DOWN ime Sequence BW f B = 8 / f B π Normalized Radian Frequency Fs = ω = he Same ime Sequence π f B π π Normalized Radian Frequency ω y[n] ime Sequence BW / Fs f B = = f B π Normalized Radian Frequency ω B Down-Sampling 9 /5/
Coarse Sequence Spectrum Linear Frequency Linear Frequency Normalized Radian Frequency / = / Normalized Radian Frequency y[n] DOWN ime Sequence / f B High Freq = / f B π ω Normalized Radian Frequency = he Same ime Sequence f B High Freq π π f B π ω y [n] ime Sequence / = / Normalized Radian Frequency f B High Freq π π f B π ω Coarse Sequence Spectrum Expanded Normalized Radian Frequency B Down-Sampling /5/
Coarse Sequence Generation x d [n] Sampling Period x d [n] Remove Zero Samples y[n] p[n] Sampling Period p[n] y[n] ime Sequence t B Down-Sampling /5/
Down Sampling in wo Steps v[n] Sampling Period Suppress D Samples v[n] Remove D Zeros & Pack y[n] δ D [n] f B Highest Frequency Sampling Period δ D [n] D y[n] ime Sequence D t B Down-Sampling /5/
Down-Sampling Operator X δ D [n] D v[n] D D t D D y[n] y[n] D-fold Decimator y[n] = S /D = x[n D] Decrease sampling frequency by a factor of /D Increase sampling period by a factor of D D = y[] = x[ ] y[] = x[ ] y[] = x[ ] y[] = x[ ] B Down-Sampling /5/
Down-Sampling Operator X δ D [n] D v[n] D D t D D y[n] y[n] D-fold Decimator y[n] = S /D = x[n D] D-fold Decimator = e j ω n π ω +π y[n] = e j ω D n π D ω D +π D stretched π D ω + π D π ω D +π w/o aliasing ω > + π D ω D > +π with aliasing B Down-Sampling /5/
Suppressing D- Samples () X δ D [n] D v[n] D D t D D δ D [n] = D k = e j π k n/ D = D D D = if mod(n, D) = e j π n e j π n/ D = otherwise (e jπ k( n/ D) = e j π k m = ) D v[n] = δ D [n] = D k= j π k n/ D e ω k = π k / D D V ( z) = D k= D V ( ω) = D k= + X (e j π k/ D z) Z {e j ω n k } = (e j ω k z) n = X (e j ω k z ) n= X ( ω π k / D) X (e j π k / D z) = X (e jω ke j ω ) = X (e j ( ω ω k) ) z = e j ω B Down-Sampling 5 /5/
Removing D- Samples and Packing () X δ D [n] D v[n] D D t D V ( z) = + v[] z + + + + v [ D] z D + + + + v[d] z D + V ( z) = + n= v [n] z n = + v[m D] z m D = F ( z D ) m= D V ( z) = D k= X (e j π k/ D z) Y (z) = V ( z / D ) = + v [] z + v[ D] z + v[d] z + v[d] z + = + y [] z + y[] z + y[] z + y [] z + + n= + v[n] z n /D = v[m D] z m m= y[n] D V ( ω) = D k= X ( ω π k / D) D Y (z ) = V ( z / D ) = D k= X (e j π k / D z /D ) D Y ( ω) = D k= X ( ω D k π D ) B Down-Sampling 6 /5/
Down-Sampling Spectra () v[n] D y[n] ime Sequence f B D X (z) D V ( z) = D k= X (e j π k/ D z) D Y (z) = D k= X (e jπ k/ D z /D ) X ( ω) D V ( ω) = D k= X ( ω π k / D) D Y ( ω) = D k= X ( ω/ D π k / D) X ( f ) V ( f ) V ( f ) X ( ω) π( / ) V ( ω) π( ) V ( ω) π( ) π π π B Down-Sampling 7 /5/
Down-Sampling Spectra () v[n] D y[n] ime Sequence f B D X (z) D V ( z) = D k= X (e j π k/ D z) D Y (z) = D k= X (e jπ k/ D z /D ) X ( ω) D V ( ω) = D k= X ( ω π k / D) D Y ( ω) = D k= X ( ω/ D π k / D) X ( f ) V ( f ) Y ( f ) X ( ω) π( / ) V ( ω) π( ) Y ( ω) π( ) π π π B Down-Sampling 8 /5/
Down Sampling v[n] y[n] ime Sequence Sampling Period Suppress D Samples v[n] Remove D Zeros & Pack y[n] V ( f ) Y ( f ) δ D [n] V ( ω) π( ) π Amplitude Axis Scaling / D Spectrum Replication / D ime Axis Scaling / D Stretching Spectrum D Y ( ω) π( ) π B Down-Sampling 9 /5/
Example When D= () δ [n] D = v[n] t n = e j ωn δ [n] = ( + ( )n ) = ( + e j π n ) (e j π = ) v [n] = x[ n] + e jπ n = e j ω n + e j π n e j ωn = e j ω n + e+ j(ω π) n V ( z) = + n= ( z 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) B Down-Sampling /5/
Example When D= () X ( ω) π( / ) π X ( ω π) π( / ) π v[n] V ( ω) π( / ) π v [n] = x[ n] + e jπ 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) B Down-Sampling /5/
Example When D= () v[n] V ( ω) π( ) π v[n] V ( ω) π( ) π B Down-Sampling /5/
Example When D= () v[n] V ( ω) π( ) π v[n] V ( ω) π( ) V ( f ) v[n] V ( ω) π π( ) π v[n] V ( ω) π( ) π B Down-Sampling /5/
Measuring Rotation Rate B Down-Sampling /5/
Signals with Harmonic Frequencies () Hz cycle / sec Hz cycles / sec Hz cycles / sec Hz cycles / sec 5 Hz 5 cycles / sec 6 Hz 6 cycles / sec 7 Hz 7 cycles / sec cos ( πt) cos ( π t) cos ( πt) cos ( 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( π)t + e j( π)t = e+ j( π)t j( 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 B Down-Sampling 5 /5/
Signals with Harmonic Frequencies () Hz cycle / sec Hz cycles / sec Hz cycles / sec Hz cycles / sec 5 Hz 5 cycles / sec 6 Hz 6 cycles / sec 7 Hz 7 cycles / sec 6 8 6 8 B Down-Sampling 6 /5/
Sampling Frequency Hz sample / sec Hz samples / sec Hz samples / sec Hz samples / sec 5 Hz 5 samples / sec 6 Hz 6 samples / sec 7 Hz 7 samples / sec 6 8 6 8 B Down-Sampling 7 /5/
Nyquist Frequency Hz cycle / sec Hz cycles / sec Hz cycles / sec Hz cycles / sec 5 Hz 5 cycles / sec 6 Hz 6 cycles / sec 7 Hz 7 cycles / sec x sample / sec x samples / sec x samples / sec x samples / sec x5 samples / sec x6 samples / sec x7 samples / sec B Down-Sampling 8 /5/
Aliasing Hz 7 Hz 6π x samples / sec π Hz 6 Hz 6π x samples / sec 6π Hz 5 Hz 6 6 6π x samples / sec 6π Hz 8 8 6π x samples / sec B Down-Sampling 9 /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 B Down-Sampling /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 = s rad = rad B Down-Sampling /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 B Down-Sampling /5/
B Down-Sampling /5/
B Down-Sampling /5/
B Down-Sampling 5 /5/
References [] http://en.wikipedia.org/ [] J.H. McClellan, et al., Signal Processing First, Pearson Prentice Hall, [] A graphical interpretation of the DF and FF, by Steve Mann [] R. Cristi, Modern Digital Signal Processing /5/