Method and apparatus for selection of ARQ parameters and estimation of improved communications

• US 6,718,493 B1
• Filed: 05/17/2001
• Issued: 04/06/2004
• Est. Priority Date: 05/17/2001
• Status: Expired due to Fees

• Current Assignee: Hewlett Packard Enterprise Development LP (Hewlett-Packard Enterprise Company)
• Original Assignee: 3Com Corporation (HP Inc.)
• Inventors: Mitlin, Vlad, Mueller, A. Joseph
• Primary Examiner(s): Ton, David
• Application Number:
  US09/859,867
• Time in Patent Office: 1,055 Days
• Field of Search: 375/200, 375/220, 375/260, 375/227, 714/774, 714/704, 714/752, 714/786, 370/252, 455/260
• US Class Current: 714/704
• CPC Class Codes: H03M 13/353 Adaptation to the channel
• CPC Class Codes: H03M 13/6306 Error control coding in combination with Automatic Repeat reQuest [ARQ] and diversity transmission, e.g. coding schemes for the multiple transmission of the same information or the transmission of incremental redundancy H03M13/3761, H03M13/3769 and H03M13/635 take precedence
• CPC Class Codes: H03M 13/6337 Error control coding in combination with channel estimation
• CPC Class Codes: H04L 1/0009 by adapting the channel coding H04L1/1812 takes precedence
• CPC Class Codes: H04L 1/1809 Selective-repeat protocols
• CPC Class Codes: H04L 1/1816 with retransmission of the same, encoded, message
• CPC Class Codes: H04L 1/1825 Adaptation of specific ARQ protocol parameters according to transmission conditions
• CPC Class Codes: H04L 1/203 Details of error rate determination, e.g. BER, FER or WER
• CPC Class Codes: H04L 27/2601 Multicarrier modulation systems

Abstract

The present invention includes a method for selecting appropriate forward error correction and automatic repeat request parameters for protection of a multicarrier communication channel. The multicarrier communication channel may be wired or wireless. Particular aspects of the present invention are described in the claims, specification and drawings.

Claims

1. A method of approximating a bit error rate applicable to a multicarrier communications channel, including:

measuring a signal to noise ratio (SNR) across the multiple subcarriers of the channel;

analytically approximating a bit error rate (BER or ε

) from at least the SNR, an average bit loading of the subcarriers (b) and forward error correction and automatic repeat request (FEC/ARQ) parameters.

2. A method of claim 1, wherein analytically approximating the BER includes applying the formula:

3. A method of claim 1, wherein analytically approximating the BER includes applying the formula:

4. A method of claim 1, wherein analytically approximating the BER includes applying the formula:

5. The method of claim 4, wherein the O term is ignored or given a small, approximate value.

6. The method of claim 1, wherein the computing step includes applying the formula:

$k_{2,\mathrm{opt}}=\frac{1+\mathrm{\Theta <br><br>}<br><br>\left({\mathrm{\gamma <br><br>}}_2\right)}{1+\mathrm{\Theta <br><br>}<br><br>\left({\mathrm{\gamma <br><br>}}_2-{\mathrm{\mu <br><br>}}_{l,\mathrm{opt},2}\right)}<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{where}$$\mathrm{\Theta <br><br>}<br><br>\left(\mathrm{\gamma <br><br>}\right)=\frac{\left(0.5<br><br>R+1\right)<br><br>\ln <br><br>\left[\begin{array}{c}\frac{4<br><br>\mathrm{\alpha <br><br>}<br><br>\left(1-2^{-b/2}\right)}{\left(3+2<br><br>b\right)}<br><br>\mathrm{erfc}<br><br>\left(\sqrt{3·<br><br>{10}^{\mathrm{\gamma <br><br>}/10}/\left(2^{b+1}-2\right)}\right)\times <br><br>\\ \left(2-\left(1-2^{-b/2}\right)<br><br>\mathrm{erfc}<br><br>\left(\sqrt{3·<br><br>{10}^{\mathrm{\gamma <br><br>}/10}/\left(2^{b+1}-2\right)}\right)\right)\end{array}\right]}{\ln <br><br>\left[\frac{\mathrm{\Gamma <br><br>}<br><br>\left({\mathrm{Sn}}_{\mathrm{eff}}<br><br>b/\mathrm{\alpha <br><br>}+1\right)}{\mathrm{\Gamma <br><br>}<br><br>\left({\mathrm{Sn}}_{\mathrm{eff}}<br><br>b/\mathrm{\alpha <br><br>}-0.5<br><br>R\right)<br><br>\mathrm{\Gamma <br><br>}<br><br>\left(0.5<br><br>R+2\right)}\right]}$

wherein

α

is the FEC/ARQ parameter for a bit size of an FEC code symbol;

γ

is the measured SNR in dB;

b is an average bit loading of effective, subcarriers of the multicarrier communication channel;

Γ

is a gamma-function;

erfc is a function $\mathrm{erfc}<br><br>\left(x\right)=\frac{2}{\sqrt{\mathrm{\pi <br><br>}}}<br><br>{\int <br><br>}_x^{\mathrm{\infty <br><br>}}<br><br>\exp <br><br>\left(-{\mathrm{\gamma <br><br>}}^2\right)<br><br>\text{\hspace{1em}<br><br>}<br><br><br><br>y;<br><br>$

and

Sn_effb=α

(K+ρ

S+R)

α

is the FEC/ARQ parameter for a bit size of an FEC code symbol;

ρ

is the FEC/ARQ parameter for a number of overhead code symbols per DMT symbol (not including the FEC control code symbols);

K is the FEC/ARQ parameter for a number of information code symbols in an FEC frame; and

S is the FEC/ARQ parameter for a number of DMT symbols per the FEC frame.

7. The method of claim 1, wherein the computing step includes applying the formula:

$\frac{k_{2,\mathrm{opt}}}{k,1,\mathrm{opt}}=\frac{1+\mathrm{\Theta <br><br>}<br><br>\left({\mathrm{\gamma <br><br>}}_2\right)}{1+\mathrm{\Theta <br><br>}<br><br>\left({\mathrm{\gamma <br><br>}}_1\right)}<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{where}$$\mathrm{\Theta <br><br>}<br><br>\left(\mathrm{\gamma <br><br>}\right)=\frac{\left(0.5<br><br>R+1\right)<br><br>\ln <br><br>\left[\begin{array}{c}\frac{4<br><br>\mathrm{\alpha <br><br>}<br><br>\left(1-2^{-b/2}\right)}{\left(3+2<br><br>b\right)}<br><br>\mathrm{erfc}<br><br>\left(\sqrt{3·<br><br>{10}^{\mathrm{\gamma <br><br>}/10}/\left(2^{b+1}-2\right)}\right)\times <br><br>\\ \left(2-\left(1-2^{-b/2}\right)<br><br>\mathrm{erfc}<br><br>\left(\sqrt{3·<br><br>{10}^{\mathrm{\gamma <br><br>}/10}/\left(2^{b+1}-2\right)}\right)\right)\end{array}\right]}{\ln <br><br>\left[\frac{\mathrm{\Gamma <br><br>}({\mathrm{Sn}}_{\mathrm{eff}}<br><br>b/\mathrm{\alpha <br><br>}+1}{\mathrm{\Gamma <br><br>}<br><br>\left({\mathrm{Sn}}_{\mathrm{eff}}<br><br>b/\mathrm{\alpha <br><br>}-0.5<br><br>R\right)<br><br>\mathrm{\Gamma <br><br>}<br><br>\left(0.5<br><br>R+2\right)}\right]}$

wherein

α

is the FEC/ARQ parameter for a bit size of an FEC code symbol;

γ

is the measured SNR in dB;

b is an average bit loading of effective subcarriers of the multicarrier communication channel;

Γ

is a gamma-function;

erfc is a function $\mathrm{erfc}<br><br>\left(x\right)=\frac{2}{\sqrt{\mathrm{\pi <br><br>}}}<br><br>{\int <br><br>}_x^{\mathrm{\infty <br><br>}}<br><br>\exp <br><br>\left(-{\mathrm{\gamma <br><br>}}^2\right)<br><br>\text{\hspace{1em}<br><br>}<br><br><br><br>y;<br><br>$

and

Sn_effb=α

(K+ρ

S+R)

α

is the FEC/ARQ parameter for a bit size of an FEC code symbol;

ρ

is the FEC/ARQ parameter for a number of overhead code symbols per DMT symbol (not including the FEC control code symbols);

K is the FEC/ARQ parameter for a number of information code symbols in an FEC frame; and

S is the FEC/ARQ parameter for a number of DMT symbols per the FEC frame.

8. A method of analytically approximating bit error rates applicable to a multicarrier communications channel, including:

selecting a range of signal to noise ratios (SNRs) applicable to the multiple sub carriers of the channel;

selecting a set of alternative communication parameters describing an average bit loading of the subcarriers (b) and forward error correction and automatic repeat request (FEC/ARQ) handling;

analytically approximating a bit error rate (BER or ε

) for the range of SNRs and the set of alternative communication parameters; and

recording the analytical approximations on machine readable memory.

9. The method of claim 8, wherein analytically approximating the BER includes applying the formula:

10. A method of claim 8, wherein analytically approximating the BER includes applying the formula:

11. A method of claim 8, wherein analytically approximating the BER includes applying the formula:

12. The method of claim 11, wherein the O term is ignored or given a small, approximate value.

13. A method of approximating a signal to noise ratio gain resulting from application of an automatic repeat request protocol (ARQ) with a particular maximum number of transmissions allowed, to a forward error correction encoded multicarrier communications channel, including:

measuring an initial signal to noise ratio across the effective subcarriers of the channel;

analytically approximating a signal to noise ratio (SNR) gain from at least the initial SNR (γ

or γ

₁), an average bit loading of the effective subcarriers (b), forward error correction (FEC) parameters and an ARQ parameter (k) for the particular maximum number of transmissions allowed.

14. The method of claim 13, wherein analytically approximating the SNR gain (μ

₁) includes applying the formula;

$k=\frac{1+\mathrm{\Theta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>\left(R,S,n_{\mathrm{eff}},b,\mathrm{\gamma <br><br>}\right)}{1+\mathrm{\Theta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>\left(R,S,n_{\mathrm{eff}},b,\mathrm{\gamma <br><br>}-{\mathrm{\mu <br><br>}}_l\right)}$

where $\mathrm{\Theta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>\left(R,S,n_{\mathrm{eff}},b,\mathrm{\gamma <br><br>}\right)=\frac{\left(0.5<br><br>R+1\right)<br><br>\text{\hspace{1em}<br><br>}<br><br>\ln <br><br>\left[\begin{array}{c}\frac{4<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{\alpha <br><br>}<br><br>\left(1-2^{b<br><br>\text{/}<br><br>2}\right)}{\left(3+2<br><br>b\right)}<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{erfc}<br><br>\text{\hspace{1em}<br><br>}<br><br>\left(\sqrt{3·<br><br>{10}^{\mathrm{\gamma <br><br>}<br><br>\text{/}<br><br>10}<br><br>\text{/}<br><br>\left(2^{b+1}-2\right)}\right)\times <br><br>\\ \left(2-\left(1-2^{-b<br><br>\text{/}<br><br>2}\right)<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{erfc}<br><br>\text{\hspace{1em}<br><br>}<br><br>\left(\sqrt{3·<br><br>{10}^{\mathrm{\gamma <br><br>}<br><br>\text{/}<br><br>10}<br><br>\text{/}<br><br>\left(2^{b+1}-2\right)}\right)\right)\end{array}\right]}{\ln <br><br>\left[\frac{\mathrm{\Gamma <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>\left({\mathrm{Sn}}_{\mathrm{eff}}<br><br>\text{\hspace{1em}<br><br>}<br><br>b/\mathrm{\alpha <br><br>}+1\right)}{\mathrm{\Gamma <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>\left({\mathrm{Sn}}_{\mathrm{eff}}<br><br>\text{\hspace{1em}<br><br>}<br><br>b<br><br>\text{/}<br><br>\mathrm{\alpha <br><br>}-0.5<br><br>R\right)<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{\Gamma <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>\left(0.5<br><br>R+2\right)}\right]}$

wherein

α

is the FEC/ARQ parameter for a bit size of an FEC code symbol;

Γ

is a gamma-function;

erfc is a function $\mathrm{erfc}<br><br>\text{\hspace{1em}<br><br>}<br><br>\left(x\right)=\frac{2}{\sqrt{\mathrm{\pi <br><br>}}}<br><br>\text{\hspace{1em}<br><br>}<br><br>{\int <br><br>}_x^{\mathrm{\infty <br><br>}}<br><br>\exp <br><br>\text{\hspace{1em}<br><br>}<br><br>\left(-y^2\right)<br><br>\text{\hspace{1em}<br><br>}<br><br><br><br>y;<br><br>$

Sn_effb=α

(K+ρ

S+R) wherein

ρ

is the FEC/ARQ parameter for a number of overhead code symbols per DMT symbol (not including the FEC control code symbols);

K is the FEC/ARQ parameter for a number of information code symbols in an FEC frame;

R is the FEC/ARQ parameter for a number of the FEC control symbols per the FEC frame; and

S is the FEC/ARQ parameter for a number of DMT symbols per the FEC frame.

15. The method of claim 13, wherein analytically approximating the SNR gain μ

₁ includes applying the formula μ

_l=10 log k.

16. The method of claim 15, further including approximating a net SNR gain μ

_n, by applying the formula;

μ

_n=μ

_l−

10 logν

where an average number of transmissions in the channel is $v=\left(1-\frac{K+\mathrm{\rho <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>S+R}{0.5·<br><br>R+1}<br><br>\text{\hspace{1em}<br><br>}<br><br>{\mathrm{\varepsilon <br><br>}}_s\right)<br><br>\text{/}<br><br>\left[1-{\left(\frac{K+\mathrm{\rho <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>S+R}{0.5·<br><br>R+1}<br><br>\text{\hspace{1em}<br><br>}<br><br>{\mathrm{\varepsilon <br><br>}}_s\right)}^{1<br><br>\text{/}<br><br>k}\right]$

wherein

ε

_s is a code symbol bit error rate for the communications channel;

ρ

is the FEC/ARQ parameter for a number of overhead code symbols per 8 DMT symbol (not including the FEC control code symbols);

K is the FEC/ARQ parameter for a number of information code symbols in an FEC frame;

k is the FEC/ARQ parameter for a maximum number of transmissions allowed; and

R is the FEC/ARQ parameter for a number of the FEC control symbols per the FEC frame;

S is the FEC/ARQ parameter for a number of DMT symbols per the FEC frame.

17. The method of claim 16, wherein the value of the code symbol bit error rate ε

_s is approximated by applying the following formula;

18. The method of claim 15, further including approximating a net SNR gain μ

_n, by applying the formula;

19. The method of claim 18, wherein the value of the bit error rate ε

is approximated by applying the following formula;

20. The method of claim 16, wherein the value of the code symbol bit error rate ε

_s is approximated by applying the following formula;

21. The method of claim 16, wherein the value of the bit error rate ε

is approximated by applying the following formula;

22. A method of approximating a signal to noise ratio gain resulting from an re-application of an automatic repeat request protocol (ARQ) with a particular maximum number of transmissions allowed, to a forward error correction encoded and ARQ protected multicarrier communications channel, including:

measuring an initial signal to noise ratio across the multiple subcarriers of the channel;

analytically approximating a signal to noise ratio (SNR) gain from at least the initial SNR (γ

or γ

₁), an average bit loading of the subcarriers (b), forward error correction (FEC) parameters and an ARQ parameters (k₁ and k₂) for the particular maximum number of transmissions allowed in the initial and modified state of the channel.

23. The method of claim 22, wherein analytically approximating the SNR gain μ

₁ includes applying the formula;

$\frac{k_2}{k_1}=\frac{1+\mathrm{\Theta <br><br>}<br><br>\left(\mathrm{\gamma <br><br>}\right)}{1+\mathrm{\Theta <br><br>}<br><br>\left(\mathrm{\gamma <br><br>}-{\mathrm{\mu <br><br>}}_l\right)}$

where $\mathrm{\Theta <br><br>}<br><br>\left(R,S,n_{\mathrm{eff}},b,\mathrm{\gamma <br><br>}\right)=\frac{\left(0.5<br><br>R+1\right)<br><br>\ln <br><br>\left[\begin{array}{c}\frac{4<br><br>\mathrm{\alpha <br><br>}<br><br>\left(1-2^{-b/2}\right)}{\left(3+2<br><br>b\right)}<br><br>\mathrm{erfc}<br><br>\left(\sqrt{3·<br><br>{10}^{\mathrm{\gamma <br><br>}/10}/\left(2^{b+1}-2\right)}\right)\times <br><br>\\ \left(2-\left(1-2^{-b/2}\right)<br><br>\mathrm{erfc}<br><br>\left(\sqrt{3·<br><br>{10}^{\mathrm{\gamma <br><br>}/10}/\left(2^{b+1}-2\right)}\right)\right)\end{array}\right]}{\ln <br><br>\left[\frac{\mathrm{\Gamma <br><br>}<br><br>\left({\mathrm{Sn}}_{\mathrm{eff}}<br><br>b/\mathrm{\alpha <br><br>}+1\right)}{\mathrm{\Gamma <br><br>}<br><br>\left({\mathrm{Sn}}_{\mathrm{eff}}<br><br>b/\mathrm{\alpha <br><br>}-0.5<br><br>R\right)<br><br>\mathrm{\Gamma <br><br>}<br><br>\left(0.5<br><br>R+2\right)}\right]}$

wherein

α

is the FEC/ARQ parameter for a bit size of an FEC code symbol;

Γ

is a gamma-function;

erfc is a function $\mathrm{erfc}<br><br>\left(x\right)=\frac{2}{\sqrt{\mathrm{\pi <br><br>}}}<br><br>{\int <br><br>}_x^{\mathrm{\infty <br><br>}}<br><br>\exp <br><br>\left(-{\mathrm{\gamma <br><br>}}^2\right)<br><br>\text{\hspace{1em}<br><br>}<br><br><br><br>y;<br><br>$

Sn_effb=α

(K+ρ

S+R) wherein

ρ

is the FEC/ARQ parameter for a number of overhead code symbols per DMT symbol (not including the FEC control code symbols);

K is the FEC/ARQ parameter for a number of information code symbols in an FEC frame;

R is the FEC/ARQ parameter for a number of the FEC control symbols per the FEC frame; and

S is the FEC/ARQ parameter for a number of DMT symbols per the FEC frame.

24. The method of claim 22, wherein analytically approximating the SNR gain μ

₁ includes applying the formula ${\mathrm{\mu <br><br>}}_l=10<br><br>\text{\hspace{1em}<br><br>}<br><br>\log <br><br>\left(\frac{k_2}{k_1}\right).$

25. The method of claim 24, further including approximating a net SNR gain μ

_n, by applying the formula;

26. The method of claim 25, wherein the value of the code symbol bit error rate ε

_s is approximated by applying the following formula;

27. The method of claim 24, further including approximating a net SNR gain μ

_n, by applying the formula;

28. The method of claim 27, wherein the value of the bit error rate ε

is approximated by applying the following formula;

29. The method of claim 25, wherein the value of the code symbol bit error rate ε

_s is approximated by applying the following formula;

30. The method of claim 25, wherein the value of the bit error rate ε

is approximated by applying the following formula;

31. A method of finding a new preferred maximum number of transmissions allowed (k_2,opt) to protect a multicarrier communication channel with forward error correction (FEC), upon a change in a base SNR (γ

₂−

γ

₁), given an initial line SNR gain due to ARQ (μ

_l,opt,1) for an initial preferred maximum number of transmissions allowed (k_l,opt,1), including;

determining a new line SNR gain (μ

_l,opt,2) as a sum of the initial line SNR gain (μ

_l,opt,1) and the change in the base SNR (γ

₂−

γ

₁); and

computing a new preferred maximum number of transmissions allowed (k_2,opt) from the new line SNR gain (μ

₂).

32. A method of evaluating a new SNR gain due to ARQ (μ

_l,opt,2) for a new SNR (γ

₂) average across effective subcarriers of a multicarrier communication channel with automatic repeat request protocol (ARQ), given an initial base SNR (γ

₁), including;

measuring the new SNR (γ

₂);

calculating or retrieving from a table an initial SNR gain due to ARQ (μ

_l,opt,1); and

computing a new SNR gain due to ARQ (μ

_l,opt,2) as μ

_l,opt,2=−

γ

₁+γ

₂.

33. The method of claim 32, wherein the communications channel further includes forward error correction (FEC).

34. A method of evaluating a net effect of a changed maximum number of transmissions allowed parameter (k) in a automatic repeat request protocol (ARQ) protecting a multicarrier communications channel with forward error correction (FEC), including:

selecting at least one signal to noise ratio (SNR or γ

) applicable to the effective subcarriers of the channel;

choosing a target SNR gain due to ARQ (μ

_l(i));

analytically approximating a number of transmissions allowed allowed that is required (k(i)) to achieve the target SNR gain and a corresponding net SNR gain (μ

_n(i));

wherein said analytically approximating step utilizes at least the selected SNR, an average bit loading of the subcarriers (b) and forward error correction and automatic repeat request (FEC/ARQ) parameters;

repeating the choosing and analytically approximating steps at least until a point of maxima is observed in the net SNR gain; and

recording on machine readable memory a result from at least one of the analytically approximating steps.

35. A method of evaluating a net effect of a changed maximum number of transmissions allowed parameter (k) in a automatic repeat request protocol (ARQ) protecting a multicarrier communications channel with forward error correction (FEC), including:

selecting at least one signal to noise ratio (SNR or γ

) applicable to the effective subcarriers of the channel;

choosing a maximum number of transmissions allowed (k(i));

analytically approximating a net SNR gain (μ

_n(i)) corresponding to the maximum number of transmissions allowed (k(i));

wherein said analytically approximating step utilizes at least the selected SNR, an average bit loading of the subcarriers (b) and forward error correction and automatic repeat request (FEC/ARQ) parameters;

repeating the choosing and analytically approximating steps at least until a point of maxima is observed in the net SNR gain; and

recording on machine readable memory a result from at least one of the analytically approximating steps.

36. A method of precalculating preferred sets of forward error correction and automatic repeat request protocol (FEC/ARQ) parameters to protect a multicarrier communication channel characterized by a number of effective subcarriers (n_eff) and an average signal to noise ratio (γ

_eff) across the effective subcarriers of the communication channel, including;

for characteristic values of n_eff and γ

_eff, using allowable FEC parameter sets, computing net SNR gain due to FEC without ARQ;

selecting for particular characteristic values of n_eff and γ

_eff, a plurality of FEC parameter sets based on said net SNR gain due to FEC;

for the selected FEC parameter sets, computing net SNR gain due to ARQ results corresponding to a plurality of maximum number of transmissions allowed (k) values; and

storing on machine readable memory, retrievable by characteristic values of n_eff and γ

_eff, computed net SNR gain due to FEC and ARQ results corresponding to at least some of the total number of transmissions (k) values and at least some of the FEC parameter sets.

37. The method of claim 36, further including for characteristic values of n_eff and γ

_eff, selecting one pair of maximum number of transmissions allowed (k) values and the FEC parameter sets, based on evaluation of at least the net SNR gain due to FEC and ARQ results.

38. A method of selecting a preferred set of forward error correction and automatic repeat request protocol (FEC/ARQ) parameters to protect a multicarrier communications channel, including:

for allowable FEC parameter sets, computing (or extracting from pre-computed data) net performance gain due to FEC without ARQ at various signal to noise ratio (SNR or y) levels;

selecting a plurality of FEC parameter sets at the SNR levels based on said net SNR gain due to FEC;

for selected FEC parameter sets at the various SNR levels, computing the corresponding net SNR gain with ARQ at the maximum number of transmissions (k); and

selecting the total number of transmissions allowed (k) and the FEC parameter set to use to protect the multicarrier communication channel based on at least the net SNR gain due FEC and the net SNR gain due to ARQ.