Up-Sampling (5B)
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= 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 ) = + n = X ( f m ) 5B Up-Sampling 3
= 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 ) = + n = X ( f m ) X ( f ) A π π + π + π ω 5B Up-Sampling 4
Increasing Sampling Frequency F s ' = UP / / 4 F s = 4 4 4 3 F s ω/ π f f 4 s f f f 4 s 4 s 4 s s 3 F s ' ω/ π Sampling Frequency F s ' = Sampling Frequency F s = 4 Sampling ime ' = Sampling ime = 4 4 4 5B Up-Sampling 5
Fine Sequence & Spectrum x[n] ime Sequence UP F s = = / / 4 4 F s = 4 = 4 4 4 Linear Frequency 3 F s ω/ π f f 4 s f f f 4 s 4 s 4 s s / F s / 4 = compressed Linear Frequency F s ' = 8 3 F s ' ω/ π ω π 3 π π π π π 3 π π Normalized Radian Frequency Normalized Radian Frequency ω 5B Up-Sampling 6
Normalized Radian Frequency x[n] ime Sequence ω = ω s = ω / s ω = ω = π f 4 f = /4 = 4 he Same ime Sequence Normalized Radian Frequency f = 4 / = 4 he Highest Frequency: y ' [n] ime Sequence 4 Normalized to Normalized Radian Frequency 5B Up-Sampling 7
Fine Sequence Spectrum Linear Frequency x[n] BW / Fs = 8 = UP ime Sequence BW = 8 /4 4 BW 4 3 4 Fs 4 = 4 4 4 4 3 4 = he Same ime Sequence BW ime Sequence BW /4 Fs 4 = = 4 4 3 4 BW 5B Up-Sampling 8
Fine Sequence Spectrum Normalized Frequency x[n] BW / Fs = 8 = UP ime Sequence BW = 8 /4 π 3 π π π Normalized Radian Frequency Fs 4 = 4 ω = 4 he Same ime Sequence π 3 π 4 π π Normalized Radian Frequency ω ime Sequence BW /4 Fs 4 = = π 3 π π 4 π Normalized Radian Frequency ω 5B Up-Sampling 9
Fine Sequence Spectrum Linear Frequency Linear Frequency Normalized Radian Frequency x[n] / = / Normalized Radian Frequency UP ime Sequence /4 4 4 High Freq 3 4 4 = 4 /4 π π 3 π π ω Normalized Radian Frequency = 4 he Same ime Sequence 4 4 High Freq 3 4 π π 3 π 4 π ω y [n] ime Sequence /4 = /4 Normalized Radian Frequency 4 4 3 4 High Freq π 3 π π 4 π ω Fine Sequence Spectrum Compressed Normalized Radian Frequency 5B Up-Sampling
Fine Sequence Generation y u [n] 4 4 4 Sampling Period Insert Zero Samples y u [n] Interpolate with Filter x[n] Sampling Period x[n] ime Sequence 5B Up-Sampling
Up Sampling in wo Steps ime Sequence v[n] 4 Highest Frequency Sampling Period 4 Interpolation Unpack & Insert D Zeros v[n] Restore D Samples x[n] Highest Frequency Sampling Period x[n] ime Sequence 5B Up-Sampling
Up-Sampling Operator v[n] D x[n] 4 D L v[n] v[n] D L-fold Expander 4 D v[n] = S L = y[n/ L] Increase sampling frequency by a factor of L if mod(n / L) = otherwise Decrease sampling period by a factor of /L n= = n= = n= =4 n=3 =6 D = v[] = y [] v[] = v[] = y[] v[3] = v[4] = y[] v[5] = v[6] = y[3] v[6] = 5B Up-Sampling 3
Up-Sampling Operator v[n] D x[n] 4 D L v[n] v[n] D L-fold Expander 4 D v[n] = S L = y[n/ L] if mod(n / L) = L-fold Expander = e j ω n π ω +π v[n] = e j ωn/ L δ L [n] π/ L ω/ L +π/ L compressed L π ω +L π π ω / L +π ω > +L π ω / L > +π 5B Up-Sampling 4
Example When D= () x[n] δ [n] D = v[n] t 4 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 n V ( z) = + ( x[n] z n + x[n]( z) n ) = 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) 5B Up-Sampling 5
Z-ransform Analysis δ D [n] = if n/d is an integer 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[md] z md = F (z D ) Sampling Period 5B Up-Sampling 6
Z-ransform Analysis δ [n] = ( + ( )n ) = ( + e j π n ) = if n/ is an integer (even) otherwise e jπ = v [n] = x[n] + e jπ n x[n] x[n] = e j ωn v [n] = e jω n + e j π n e j ωn = e j ωn + e+ j(ω π) n V ( z) = + ( x[n] z n + x[n]( z) n ) = n= X ( z) + X ( z) V (ω) = V (e jω ) = X (e j ω ) + X (e jπ e j ω ) = X (ω) + X (ω π) 5B Up-Sampling 7
Measuring Rotation Rate 5B Up-Sampling 8
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 5B Up-Sampling 9
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 5B Up-Sampling
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 5B Up-Sampling
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 5B Up-Sampling
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 5B Up-Sampling 3
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 5B Up-Sampling 4
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 5B Up-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 5B Up-Sampling 6
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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