Undersampling (2B) Young Won Lim 4/4/12
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1 Undersapling (2)
2 Copyright (c) Young W. Li. Perission is granted to copy, distribute and/or odify this docuent under the ters of the GNU Free Docuentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no ac-cover Texts. A copy of the license is included in the section entitled "GNU Free Docuentation License". Please send corrections (or suggestions) to youngwli@hotail.co. This docuent was produced by using OpenOffice and Octave.
3 and-liited Signal = 5 MHz = 5 MHz = 5 MHz = 5 MHz MHz MHz irror Lowpass Sapling = 5 MHz = 5 MHz andpass Sapling IF filtering Haronic Sapling Sub-Nyquist Sapling MHz Undersapling 3
4 Low-pass Signal Sapling = 5 MHz = 5 MHz MHz MHz Nyquist Criterion + 2 Undersapling 4
5 and-pass Signal Sapling = 5 MHz = 5 MHz MHz MHz irror andpass Sapling IF filtering Haronic Sapling Sub-Nyquist Sapling MHz Nyquist Criterion MHz 2 Undersapling 5
6 Sapling Frequency (1) Assue there are ultiples of Given an integer = Max condition can be decreased according to the following condition without introducing aliasing probles + = (+1) Min condition Given and-pass Signal is characterized by andwidth Carrier Frequency + / Undersapling 6
7 Sapling Frequency (2) ( + 1) Given and-pass Signal is characterized by andwidth 2 ( + 1) 2 f L Carrier Frequency Noralization by = + /2 f L = /2 Highest frequency Lowest frequency + + /2 /2 + f L f L 2 Undersapling 7
8 Exaple =6 (1) When = 6 /2 + in + 7 ax 6 6, ax ax = 6 7, in in = Undersapling 8
9 Exaple =6 (2) = 6 ax = 6 / MHz 1 < 6 1 > 6 / MHz = + 7 in + = 7 + / MHz 2 Undersapling 9
10 Exaple =6 (3) = 6 ax = 6 /2 MHz < 6 MHz = + 7 in + = 7 MHz + /2 2 Undersapling 10
11 Miniu Plot (1) X-Y Plot + /2 = R X highest signal frequency bandwidth Y,in This plot shows in noralized by, for the given bandpass signal that is characterized by R and the given paraeter + ( + 1) =, in iniu sapling rate bandwidth Y Characterized by andwidth R = X Carrier Frequency = + /2 2 Undersapling 11
12 Miniu Plot (2) = + /2 = 1 R = / = 1 X-Y Plot = + /2 = 2 R = / = 2,in Y This plot shows in noralized by, for the given bandpass signal that is characterized by R and the given paraeter = + /2 = 3 R = / = 3 X Characterized by andwidth R = Carrier Frequency = + /2 2 Undersapling 12
13 Miniu Plot (3) g (, R) = 2 (+1) = 2 (+1) R = X + /2 = R = 0 = 1 g (0, R) = 2 R g (1, R) = R slope = 2 slope = 1, in = Y + ( + 1) = 2 ( + 1) = 2 = 3 g (2, R) = 2 R 3 g (3, R) = 1 R 2 slope = 2/3 slope = 1/2 g (, R) = 4 g (4, R) = 2 5 R slope = 2/5 + = 5 g (5, R) = 1 3 R slope = 1/3 = 6 g (6, R) = 2 7 R slope = 2/7 = 7 g (7, R) = 1 4 R slope = 1/ 4 f L f L = 8 g (8, R) = 2 9 R slope = 2/9 2 Undersapling 13
14 Miniu Plot (4) g (, R) = 2 (+1) = 2 (+1) R R = +1 g (, +1) = 2 = 0 g (0, R) = 2 R slope = 2 = 0 R = 1 g (0,1) = 2 = 1 g (1, R) = R slope = 1 = 1 R = 2 g (1, 2) = 2 = 2 g (2, R) = 2 3 R slope = 2/3 = 2 R = 3 g (2, 3) = 2 = 3 g (3, R) = 1 2 R slope = 1/2 = 3 R = 4 g (3, 4) = 2 = 4 g (4, R) = 2 5 R slope = 2/5 = 4 R = 5 g (4,5) = 2 = 5 g (5, R) = 1 3 R slope = 1/3 = 5 R = 6 g (5,6) = 2 = 6 g (6, R) = 2 7 R slope = 2/7 = 6 R = 7 g (6,7) = 2 = 7 g (7, R) = 1 4 R slope = 1/ 4 = 7 R = 8 g (7,8) = 2 = 8 g (8, R) = 2 9 R slope = 2/9 = 8 R = 9 g (8,9) = 2 2 Undersapling 14
15 Miniu Plot (5),in = 1 ( = 0) = 2 ( = 1) = 3 ( = 2) slope = = slope = 1 slope = 2/3 slope = 1/2 slope = 2/5 + ( + 1) =, in = g (, R) iniu sapling rate bandwidth 1 + /2 = R highest signal frequency R = bandwidth 2 Undersapling 15
16 Min, Max Condition on (1) / = ax in ax, ax 2 2 f L f L + f L in, in + = 2 = 3 = 4 2 f L 2 3 f L f L = 1 = 2 = 3 2 Undersapling 16
17 Min, Max Condition on (2) in ax 2 2 f L /2 = ax represents how any are in 2 in ax, ax ax = = 2 f L f L + f L in, in represents how any are in 2 + in in in = + = Undersapling 17
18 Exaple =1 (=0) = 1 ( = 0) = + /2 = 1 R = / = 1,in,ax = 2 = 2 = 2( ) ( 1) = + R [1, 2] (, ) = (2, 4) = 1 ( = 0) = + /2 = 1.5 R = / = 1.5,in = 2 = 3 slope = 2,ax = 2( ) ( 1) = + (, ) = (1, 2) = 1 ( = 0) = + /2 = 2 R = / = 2,in = 2 = 4,ax = 2( ) ( 1) = + 2 Undersapling 18
19 Exaple =2 (=1) = 2 ( = 1) = 2 ( = 1) = + /2 = 2 = + /2 = 2.5 R = / = 2,in,ax = 2 = 2 = 2( ) ( 1) = 2 R = / = 2.5,in,ax = 2 = 2.5 = 2( ) ( 1) = 3 R [2, 3] (3, 4) (, ) = (3, 3) slope = 1 (, ) = (2, 2) = 2 ( = 1) = + /2 = 3 R = / = 3,in = 2 = 3,ax = 2( ) ( 1) = 4 2 Undersapling 19
20 Exaple =3 (=2) = 3 ( = 2) = + /2 = 3 R = / = 3 R [3, 4],in,ax = 2 = 2 = 2( ) ( 1) = 2 = 3 ( = 2) = + /2 = 3.5 R = / = 3.5,in = 2 = 7 3 (3, 4) (, ) = (4, 8 3 ),ax = 2( ) ( 1) = 3 slope = 2 3 (, ) = (3, 2) = 3 ( = 2) = + /2 = 4 R = / = 4,in = 2 = 8 3,ax = 2( ) ( 1) = 4 2 Undersapling 20
21 Min, Max Condition on (2) / = ax in ax, ax 2 2 f L f L + f L in, in + = 2 = 3 = 4 2 f L 2 3 f L f L = 1 = 2 = 3 2 Undersapling 21
22 Min Max Plot (1),in = 1 ( = 0) = 2 ( = 1) = 3 ( = 2) = f L R = 2 Undersapling 22
23 Min Max Plot (2),in = 1 ( = 0) = 2 ( = 1) = 3 ( = 2) f L 4 Min Max 3 = 2 2 f L 2 = f L = 2 = 3 1 [2, 3 ] [, 2( )] [3, 4 ] [ 2 3, ( )] Min y = 1( x 2)+2 y = x = 2 Max y = 2( x 2)+2 y = 2 x R = y = 2 (x 3)+2 3 = 3 y = 2 x 3 y = 1( x 3)+2 y = x 1 2 Undersapling 23
24 Range of when R=4.5, =5 (1) For a given Nyquist Criterion 2 = 20 MHz = 5 MHz in ax Optiu Sapling Frequency = = = 35 = 22.5 MHz (10 ) = = = 17.5 = 17.5 MHz (10 ) = = = = MHz (10 ) = = = 8.75 X = = = 7.0 X 2 Undersapling 24
25 Range of when R=4.5, =5 (2) For a given Nyquist Criterion 2 = 20 MHz = 5 MHz 2 2 f L = +/2 = 22.5 MHz f L = /2 = 17.5 MHz in ax = 2 = 1 2 f L = 3 = f L = 4 = f L = 5 = f L 9.0 X 8.75 = 6 = f L 7.5 Xf s Undersapling 25
26 Range of when R=4.5, =5 (3) When = 6 /2 + in + 7 ax 6 6, ax ax = 6 7, in in = Undersapling 26
27 Range of when R=4.5, =5 (4) =5 =5 2 2 f L in = 2 = 1 ax 22.5 = 2 f L = 35 Optiu = 2 = 2 =22.5 <22.5 = 1 = 1 =35 >35 2 Undersapling 27
28 Range of when R=4.5, =5 (5) =5 =5 2 2 f L in 15.0 = = 3 = 2 ax 2 = f L = 3 = 3 =15.0 <15.0 Optiu = 2 = 2 =17.5 > Undersapling 28
29 Range of when R=4.5, =5 (6) =5 =5 2 2 f L in 11.2 = = 4 = 3 ax 1 = f L Optiu = 4 = 4 =11.2 <11.2 = 3 = 3 =11.67 > Undersapling 29
30 Range of when R=4.5, =5 (7),in = 1 ( = 0) = 2 ( = 1) = 3 ( = 2) = = = =2.33 =2.24 = 2 = = 3 = = 4 = R = = =4.5 2 Undersapling 30
31 Spectral Inversion =5 =5 Optiu = 2 inverted Optiu =22.5 = 2 Optiu = 4 =17.5 = Undersapling 31
32 References [1] [2] J.H. McClellan, et al., Signal Processing First, Pearson Prentice Hall, 2003 [3] A graphical interpretation of the DFT and FFT, by Steve Mann [4] R. G. Lyons, Understanding Digital Signal Processing, 1997
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