From f695c9d55409f7d6109275ea8938fe8fd63be892 Mon Sep 17 00:00:00 2001 From: Peter Tanner Date: Sat, 2 Nov 2024 20:30:13 +0800 Subject: [PATCH] Fix sampling function (scale $f_s$) --- README.html | 12 ++++++++---- README.md | 11 +++++++++-- README.pdf | Bin 664527 -> 665102 bytes 3 files changed, 17 insertions(+), 6 deletions(-) diff --git a/README.html b/README.html index db8d14c..31cca24 100644 --- a/README.html +++ b/README.html @@ -675,14 +675,15 @@ Use these formulas in particular:

Sampling

-

t=nTsTs=1fsxs(t)=x(t)δs(t)=x(t)nZδ(tnTs)=nZx(nTs)δ(tnTs)Xs(f)=X(f)nZδ(fnTs)=X(f)nZδ(fnfs)B>12fs,2B>fsAliasing\begin{align*} +

t=nTsTs=1fsxs(t)=x(t)δs(t)=x(t)nZδ(tnTs)=nZx(nTs)δ(tnTs)Xs(f)=fsX(f)nZδ(fnTs)=fsX(f)nZδ(fnfs)    Xs(f)=nZfsX(fnfs)Sampling (FT)B>12fs    2B>fsAliasing\begin{align*} t&=nT_s\\ T_s&=\frac{1}{f_s}\\ x_s(t)&=x(t)\delta_s(t)=x(t)\sum_{n\in\mathbb{Z}}\delta(t-nT_s)=\sum_{n\in\mathbb{Z}}x(nT_s)\delta(t-nT_s)\\ - X_s(f)&=X(f)*\sum_{n\in\mathbb{Z}}\delta\left(f-\frac{n}{T_s}\right)=X(f)*\sum_{n\in\mathbb{Z}}\delta\left(f-n f_s\right)\\ - B&>\frac{1}{2}f_s, 2B>f_s\rightarrow\text{Aliasing}\\ + X_s(f)&=f_s X(f)*\sum_{n\in\mathbb{Z}}\delta\left(f-\frac{n}{T_s}\right)=f_s X(f)*\sum_{n\in\mathbb{Z}}\delta\left(f-n f_s\right)\\ + \implies X_s(f)&=\sum_{n\in\mathbb{Z}}f_s X\left(f-n f_s\right)\quad\text{Sampling (FT)}\\ + B&>\frac{1}{2}f_s\implies 2B>f_s\rightarrow\text{Aliasing}\\ \end{align*} -

+

Procedure to reconstruct sampled signal

Analog signal x(t)x'(t) which can be reconstructed from a sampled signal xs(t)x_s(t): Put xs(t)x_s(t) through LPF with maximum frequency of fs/2f_s/2 and minimum frequency of fs/2-f_s/2. Anything outside of the BPF will be attenuated, therefore nn which results in frequencies outside the BPF will evaluate to 00 and can be ignored.

@@ -708,6 +709,9 @@ Use these formulas in particular:

Nyquist criterion for zero-ISI

Do not transmit more than 2B2B samples per second over a channel of BB bandwidth.

+

Nyquist rate=2BNyquist interval=12B\text{Nyquist rate} = 2B\quad\text{Nyquist interval}=\frac{1}{2B} +

+

By Bob K - Own work, CC0, https://commons.wikimedia.org/w/index.php?curid=94674142

Insert here figure 8.3 from M F Mesiya - Contemporary Communication Systems (Add image to images/sampling.png)

Cannot add directly due to copyright!

diff --git a/README.md b/README.md index 7bcf59e..a019b65 100644 --- a/README.md +++ b/README.md @@ -490,8 +490,9 @@ Use these formulas in particular: t&=nT_s\\ T_s&=\frac{1}{f_s}\\ x_s(t)&=x(t)\delta_s(t)=x(t)\sum_{n\in\mathbb{Z}}\delta(t-nT_s)=\sum_{n\in\mathbb{Z}}x(nT_s)\delta(t-nT_s)\\ - X_s(f)&=X(f)*\sum_{n\in\mathbb{Z}}\delta\left(f-\frac{n}{T_s}\right)=X(f)*\sum_{n\in\mathbb{Z}}\delta\left(f-n f_s\right)\\ - B&>\frac{1}{2}f_s, 2B>f_s\rightarrow\text{Aliasing}\\ + X_s(f)&=f_s X(f)*\sum_{n\in\mathbb{Z}}\delta\left(f-\frac{n}{T_s}\right)=f_s X(f)*\sum_{n\in\mathbb{Z}}\delta\left(f-n f_s\right)\\ + \implies X_s(f)&=\sum_{n\in\mathbb{Z}}f_s X\left(f-n f_s\right)\quad\text{Sampling (FT)}\\ + B&>\frac{1}{2}f_s\implies 2B>f_s\rightarrow\text{Aliasing}\\ \end{align*} ``` @@ -542,6 +543,12 @@ Calculate $C_n$ coefficient as follows from $x_p(t)$: Do not transmit more than $2B$ samples per second over a channel of $B$ bandwidth. +```math +\text{Nyquist rate} = 2B\quad\text{Nyquist interval}=\frac{1}{2B} +``` + + + ![By Bob K - Own work, CC0, https://commons.wikimedia.org/w/index.php?curid=94674142](images/Nyquist_frequency_&_rate.svg) ### Insert here figure 8.3 from M F Mesiya - Contemporary Communication Systems (Add image to `images/sampling.png`) diff --git a/README.pdf b/README.pdf index 8a97b3cb9518e22c5495b83af49edce6fdf2b793..ed99660269847420abfc92c2cd1b8d1822699766 100644 GIT binary patch delta 13176 zcmV-;Gl$I2$0&}>D6lQef0G+GcKxnj(H}V#PaD9+)0K6V?Oks+m3W<9rMwksKhQHA ziP;%a=5W^Xzt1^nKAP+YNl>cTC21aP0QcfP&V2x_D~11WHU5g$RCiCWetGp@)i1AX zsEtmQnA#|*oYJ*(u2QvlsaCV9cb5%3^h`5y z>nY8rzJ2{l>)OVk@N7fm?puy%l)@m!*J7+U)$6BMUq0OGTUF`m^~0+-lOLAz=UMe( zwz|1BYVxpLJxv#Pvzx!a{`Xg*_AV+d&9J-g_Mkdo9i&;Hj@<>;7JuSdi`3Xw?GvSMY?VLW zNSG{tovps8W{cVC-EUY{HC^0S)0g$~Zn;{`^6I(H_A1Y3f0jvDwf5enk#$V!o7eyQ z>Q;0eT+mh^RJBX|uX=sI;pO`Mtl=n@^YrpKeXiz<>ce!kp5MLTM~~ChS9yhJNGDcM zyC^E8WBHu?b^7$-alUxR^`zPbY z=NUf46(275{6&fKwN~DqV9r<7k1y+kDFix|IYQkkf8+?psGW{BjZY4#UoBtWy|4b` 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