FPGA Central

Gardner TED for BPSk is e(r)=y(r-1/2)[y(r)-y(r-1)].that mean the
e(r) is related to y's amplitude and so related to
signal's power. But TED is attend to give a phasic error,which is
in[0,2*pi] and unrelated to signal's power. How
can I get the e(r) in zone [0,2*pi]? What's the normalization method?

On Jun 25, 9:17 am, peugeot888 <peugeot...@tom.com> wrote:
> Gardner TED for BPSk is e(r)=y(r-1/2)[y(r)-y(r-1)].that mean the
> e(r) is related to y's amplitude and so related to
> signal's power. But TED is attend to give a phasic error,which is
> in[0,2*pi] and unrelated to signal's power. How
> can I get the e(r) in zone [0,2*pi]? What's the normalization method?

It's intended to give *delay* estimate, not phase estimate.

On Jun 25, 10:17 am, peugeot888 <peugeot...@tom.com> wrote:
> Gardner TED for BPSk is e(r)=y(r-1/2)[y(r)-y(r-1)].that mean the
> e(r) is related to y's amplitude and so related to
> signal's power. But TED is attend to give a phasic error,which is
> in[0,2*pi] and unrelated to signal's power. How
> can I get the e(r) in zone [0,2*pi]? What's the normalization method?


Think of the TED

On Jun 25, 10:17 am, peugeot888 <peugeot...@tom.com> wrote:
> Gardner TED for BPSk is e(r)=y(r-1/2)[y(r)-y(r-1)].that mean the
> e(r) is related to y's amplitude and so related to
> signal's power. But TED is attend to give a phasic error,which is
> in[0,2*pi] and unrelated to signal's power. How
> can I get the e(r) in zone [0,2*pi]? What's the normalization method?

Think of the TED as a "timing delay to number" converter, and the NCD
(numerically-controlled delay) that follows the loop filter as a
"number to timing delay" converter ...

But what is unit for e(r)? How can I express the delay?

"BERT" <callmevc@gmail.com>
??????:1182783396.990637.224230@q75g2000hsh.google groups.com...
> On Jun 25, 10:17 am, peugeot888 <peugeot...@tom.com> wrote:
>> Gardner TED for BPSk is e(r)=y(r-1/2)[y(r)-y(r-1)].that mean the
>> e(r) is related to y's amplitude and so related to
>> signal's power. But TED is attend to give a phasic error,which is
>> in[0,2*pi] and unrelated to signal's power. How
>> can I get the e(r) in zone [0,2*pi]? What's the normalization method?
>
> Think of the TED as a "timing delay to number" converter, and the NCD
> (numerically-controlled delay) that follows the loop filter as a
> "number to timing delay" converter ...
>

You need to do some homework and derive it, there's a
dependence on the pulse shape that you use in your
modulation. It is commonly called the "gain" of the TED,
and it can be derived from the S-curve, which is the expected
output of the TED as a function of the true delay.

Do you have references on the terms I used above?


On Jun 26, 8:50 am, "peugeot888" <peugeot...@tom.com> wrote:
> But what is unit for e(r)? How can I express the delay?
>
>

julius wrote:
> You need to do some homework and derive it, there's a
> dependence on the pulse shape that you use in your
> modulation. It is commonly called the "gain" of the TED,
> and it can be derived from the S-curve, which is the expected
> output of the TED as a function of the true delay.
>
> Do you have references on the terms I used above?

For pulse shape think excess bandwidth. Gardner works great with large
excess bandwidth, and falls apart for very limited excess bandwidth.

In most practical cases, the result of the Gardner calculation should be
thought of as little more than a "go left" or "go right" indication.
Trying to use the actual magnitude of the result, rather than just its
sign, works fine in a sterile noiseless model. In the real noisy world
the impairments mean trying to scale the adjustment with the magnitude
of the result can be less than spectacularly successful.

Steve

> In most practical cases, the result of the Gardner calculation should be
> thought of as little more than a "go left" or "go right" indication.
> Trying to use the actual magnitude of the result, rather than just its
> sign, works fine in a sterile noiseless model. In the real noisy world
> the impairments mean trying to scale the adjustment with the magnitude
> of the result can be less than spectacularly successful.
>

And, that is also why the Gardener TED works for a wide range of
modulation schemes including QAM. Anyways, you might want to consult
"Digital Communication Receivers" by H. Meyr et. al, for a more in-
depth treatment of this subject.

Vijay.

What references give the definition of S-curve?I have look up
<Synchronization Techniques for Digital Receivers> . <Digital Communication
Receivers Synchronization, Channel Estimation, and Signal Processing>
and <digital communications>.But I can't find the definition of S-curve.
Could tell me the reference?thanks

"julius" <juliusk@gmail.com>
??????:1182877605.962511.200660@q69g2000hsb.google groups.com...
> You need to do some homework and derive it, there's a
> dependence on the pulse shape that you use in your
> modulation. It is commonly called the "gain" of the TED,
> and it can be derived from the S-curve, which is the expected
> output of the TED as a function of the true delay.
>
> Do you have references on the terms I used above?
>
>
> On Jun 26, 8:50 am, "peugeot888" <peugeot...@tom.com> wrote:
>> But what is unit for e(r)? How can I express the delay?
>>
>>
>
>

On Jun 27, 7:29 am, "peugeot888" <peugeot...@tom.com> wrote:
> What references give the definition of S-curve?I have look up
> <Synchronization Techniques for Digital Receivers> . <Digital Communication
> Receivers Synchronization, Channel Estimation, and Signal Processing>
> and <digital communications>.But I can't find the definition of S-curve.
> Could tell me the reference?thanks
>

Look again, it's definitely in d'Andrea and Mengali's "Synchronization
Techniques
for Digital Receivers." You didn't give the authors names, but I
assume that
you were referring to the same book.

Julius

On Jun 26, 9:41 pm, Steve Underwood <ste...@dis.org> wrote:

> In most practical cases, the result of the Gardner calculation should be
> thought of as little more than a "go left" or "go right" indication.
> Trying to use the actual magnitude of the result, rather than just its
> sign, works fine in a sterile noiseless model. In the real noisy world
> the impairments mean trying to scale the adjustment with the magnitude
> of the result can be less than spectacularly successful.
>
> Steve

This may work in some cases, but the safest approach is to derive the
"gain"
factor of the TED. This is especially important if the output is fed
into a
feedback tracking loop, since it can/will determine the performance/
stability
of the loop itself :-P.

Julius

julius wrote:
> On Jun 26, 9:41 pm, Steve Underwood <ste...@dis.org> wrote:
>
>> In most practical cases, the result of the Gardner calculation should be
>> thought of as little more than a "go left" or "go right" indication.
>> Trying to use the actual magnitude of the result, rather than just its
>> sign, works fine in a sterile noiseless model. In the real noisy world
>> the impairments mean trying to scale the adjustment with the magnitude
>> of the result can be less than spectacularly successful.
>>
>> Steve
>
> This may work in some cases, but the safest approach is to derive the
> "gain"
> factor of the TED. This is especially important if the output is fed
> into a
> feedback tracking loop, since it can/will determine the performance/
> stability
> of the loop itself :-P.
>
> Julius
>
As someone else pointed out, if you use Gardner for QAM, you can do
nothing more than use it as a nudge left/nudge right indicator. As you
say, a stability analysis is important. With limited excess bandwidth
the darned thing can get stuck in the wrong place, unless you take care.

Steve

I have read some reference and do simulation. But I can't get a
convergence
result. I attach my m-file programme and data. Who can tell me what
the
problem with my programme?


clc;
clear all;
close all;

fs=20; %fixed sample frequency of receiver
Ts=1/fs; %fixed sample period of receiver
fb=3; %symbol frequency
Tb=1/fb; %symbol period
Ti=Tb/2; %

%% parameter for loop filter
zeta = 0.707; % Damping factor

Wl =0.05*fb; % Lock range defined according to expected input
characteristics
Wn = Wl/2/zeta; % Natural frequency calculated from lock range

Kd = 2.27 % Phase detector gain
Ko = 1; % Loop filter gain

C1=((8*zeta*Wn*Tb)/(4+4*zeta*Wn*Tb+Wn*Wn*Tb*Tb))/(Ko*Kd);
C2=((4*Wn*Wn*Ts*Tb)/(4+4*zeta*Wn*Tb+Wn*Wn*Tb*Tb))/(Ko*Kd);


%%

x=importdata('basebandSample.txt');
figure;
plot(x);
title('waveform of baseband sample ');
hold on;
plot(x,'r*');

%%

uk=0; %fractional interval
mkDelta =floor(Ti/Ts+uk ); %increment of basepoint index

curMk=3; %current basepoint index

IdotCnt=1; %output index for interpolator
VdCnt=1; %output index for TED

demodOutCnt=1; %demod decision output index


Discriminator_Out(1)=0; %output of Gardner TED
PLL_Phase_Part(1)=0; %gain branch of loop filter
PLL_Freq_Part(1)=0; %integral branch of loop filter

Control(1)=0; %error value from loop filter

while curMk <= length(x)-1

y(IdotCnt)=(1-uk)*x(curMk)+uk*x(curMk+1); %linear interpolate


if IdotCnt>= 3 && (mod(IdotCnt ,2)==1)

Discriminator_Out(VdCnt)=y(IdotCnt-1)*[sign(y(IdotCnt))-
sign(y(IdotCnt-2))]; %Gardner TED for BPSK
%Discriminator_Out(VdCnt)=y(IdotCnt-1)*[y(IdotCnt)-
y(IdotCnt-2)];

demodOut(demodOutCnt)=y(IdotCnt);
demodOutCnt=demodOutCnt+1;

%loop filter
if VdCnt >= 2

PLL_Phase_Part(VdCnt)=Discriminator_Out(VdCnt)*C1;
Control(VdCnt)=PLL_Phase_Part(VdCnt)
+PLL_Freq_Part(VdCnt-1);

PLL_Freq_Part(VdCnt)=Discriminator_Out(VdCnt)*C2+P LL_Freq_Part(VdCnt-1);


%get next increment of basepoint and fractional interval
mkDelta=floor( Ti/Ts+ Control(VdCnt)+uk );
uk=mod( Ti/Ts+ Control(VdCnt)+uk ,1);

end

VdCnt=VdCnt+1;

end

curMk=curMk+mkDelta; %next basepoint
IdotCnt=IdotCnt+1;

end


figure;
plot(Discriminator_Out);

figure;
plot(Control);

scatterplot(x);
scatterplot(demodOut(80:length(demodOut)));

This is my baseband data. Generated by systemview.

1.75371715478675e-003
2.12139994902895e-002
-6.24083964762967e-002
5.21719767698147e-001
1.06779003472277e+000
9.75178281677328e-001
1.00000000000000e+000
9.66713585266056e-001
1.02784745200005e+000
9.59143375337561e-001
-5.38732211865683e-001
-1.12428879291353e+000
-9.37152879725854e-001
-1.00000000000000e+000
-1.00000000000000e+000
-1.00000000000000e+000
-1.00000000000000e+000
-9.66713585266056e-001
-1.02784745200005e+000
-9.59143375337561e-001
5.35224777556109e-001
1.08186079393295e+000
1.06196967267845e+000
-4.34395353962934e-002
-1.13558006944555e+000
-9.50356563354656e-001
-1.00000000000000e+000
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-1.00000000000000e+000
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-1.12481679295259e+000
4.34395353962934e-002
1.13558006944555e+000
9.50356563354656e-001
1.00000000000000e+000
9.66713585266056e-001
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9.57572001019421e-001
1.12481679295259e+000
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-1.00000000000000e+000
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