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Old 2-Dec-2013, 8:07 PM   #1
mmbridges
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Noise Figure Accounting

I am trying to understand how the addition of a low noise figure (NF) preamp can improve the overall system NF.

I have a Zenith DTT-901 Set Top Box (STB) that I cannot find the NF for so it so lets assume it is 6dB.

What is the formula that shows how the gain and NF of a preamp improve the overall system NF when placed in front of the STB. (this assumes the preamp NF is lower than the STB NF)

I have read that as long as the preamp gain (dB) is sufficiently larger than the STB NF (dB) then the over all system NF (dB) will be determined by the pre-amp NF (dB).

I am an equations kind of guy so would like to see exactly how as the pre-amp gain increases the system NF approaches the pre-amp NF.

Thanks!
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Old 2-Dec-2013, 10:59 PM   #2
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Have you seen this thread yet? http://forum.tvfool.com/showthread.php?t=109
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Old 2-Dec-2013, 11:40 PM   #3
mmbridges
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Missing Details

Thanks for the reply Gmast. I have seen this thread. In particular:

Scenario 3 (rooftop antenna followed by pre-amp)


indicates

(Note that his scenario assumes that the gain of the pre-amp is more than enough to overcome the losses of the cable, splitter, and receiver noise figure that come after it in the chain.)


the example suggests for preamp gain that is "more than enough", all of the above losses are cancelled out. It is the "more than enough" part I am interested in getting more details on.

Is the degree to which these losses + NF are overcome a function of the preamp gain? If so what is the formula? In other words if the preamp gain exactly equaled the referenced losses + receiver NF, would the net NF exactly equal to the preamp NF? If not what is the formula that determines how different it would be?
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Old 3-Dec-2013, 6:45 AM   #4
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You can estimate the losses in your cabling and splitter(s). Worst case, RG-6 should have no more than 6 dB loss per 100' at the top end of the UHF band. 2-way splitters should not have more than 4 dB insertion loss, 4-way splitters should not have more than 8 dB insertions loss and 8-way splitters should have no more than 12 dB insertion loss. Many splitters will be labeled, indicating loss at each port.

Simply add the length of cables starting at the antenna, all the way through to the tuner you are estimating distribution losses for, then (Total Cable Length / 100') * 6dB + Insertion Loss of the Splitter(s) in the Path = Estimated Antenna to Tuner Distribution System Loss

Given that the base thermal noise power of a 6 MHz channel is -106 dBm (http://en.wikipedia.org/wiki/Johnson...3Nyquist_noise), we can estimate the equivalent power level of the noise in a tuner if we know it's noise figure... -106 dBm + Tuner NF. In your case, assuming an estimated NF of 6 dB, the equivalent baseline noise level in the tuner is -100 dBm.

The antenna has a baseline noise level of -106 dBm for the same 6 MHz channel, so if a preamplifier is placed in the system, close to the antenna, both the signals and noise received along with the baseline noise of the antenna will be amplified. For a 20 dB gain amplifier with a 3 dB NF, we would expect signal + received noise to be increased by 20 dB, and we would also expect the baseline noise of the antenna to be elevated to a power level of -106 dBm + 20 dB = -86 dBm

Presuming that the preamplifier has a lower noise figure than the tuner... As long as the loss in the Antenna to Tuner Distribution System does not attenuate the amplified baseline noise level of the antenna to a level equal to or less than the equivalent baseline noise level in the tuner, the preamplifier NF dominates the net system NF.

In other words, if we had 14 dB of Antenna to Tuner Distribution System loss, the power level of the amplified baseline noise of the antenna would be attenuated to -86 dBm - 14 dB = -100 dBm which is equal to the tuner equivalent baseline noise level in the tuner of -100 dBm -- which would make the superior amplifier NF of no benefit. If we could reduce the distribution losses by 3 dB or more, the low NF of the preamp would be at the threshold of fully dominating the net NF of the system.
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Last edited by GroundUrMast; 3-Dec-2013 at 6:47 AM. Reason: sp.
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Old 3-Dec-2013, 12:10 PM   #5
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If you want to calculate the actual numbers, use a system "cascade noise" calculator. There are plenty of them on the internet.
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Old 4-Dec-2013, 1:48 AM   #6
mmbridges
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So it looks like Frii's formula is what I was looking for. See

http://en.wikipedia.org/wiki/Noise_figure

I believe this is the formula behind the various online cascaded noise figure calculators ADTech mentioned. See

http://www.changpuak.ch/electronics/calc_01.php

=========================================================
DISCLAIMER: What follows below will probably not be interesting to most but it helped me better understand how people were explaining noise figure accounting with respect to preamps, cable losses and tuner 1st circuit noise. What was confusing me was how various gain and noise figures were being added an subtracted sometimes but then later completely ignored using justifications like "the preamp gain is large enough". Hopefully I got it right. With luck there is maybe one other person that might find this was not a waste of time reading. I also suspect there is probably a much more succinct, easier and clearer way to explain it. If I made a mistake in my explanation please straighten me out.
=========================================================

Here is my understanding from a mathematical point of view using a simplified system consiting of three cascaded stages consisting of preamp (pa), lumped cable losses (c) and set top box 1st transistor circuit (stb).


(1) Let F_sys = system noise factor = SNR_in/SNR_out

where SNR_in = signal to noise ratio at input to preamp immediately following antenna and
SNR_out = signal to noise ratio immediately following 1st transistor circuit of set top box

Since F_sys, SNR_in and SNR_out are linear ratios we can convert to dB and write

(2) NF_sys(dB) = System Noise Figure = 10 log(F_sys) = SNR_in(dB) - SNR_out(dB)

I believe ultimately, the SNR_out(dB) has to be at least > 15dB in order for the STB decoding circuitry to lock on/decode the signal.

Lets assume the antenna location and antenna gain are such that at the preamp input, SNR_in = 18dB. We can then write

(3) NM(dB) = Noise Margin = SNR_out(dB) -15dB= SNR_in(dB) - NF_sys(dB) - 15dB = 3dB - NF_sys(dB)


Now lets look at Frii's formula for the above three stage system. Basically used the Wikipedia link above to write the following equation:


(4) F_sys = F_pa + (F_c-1)/A_pa + (F_stb -1)/(A_pa*A_c)

where:

G_pa = preamp gain (dB) > = 0
L_c = cable loss (db) < = 0
NF_pa = preamp noise figure (dB) > = 0
NF_c = equivalent cable loss noise figure (dB) = -L_c > = 0
NF_stb = STB 1st circuit noise figure (dB) > = 0


F_pa = preamp noise factor (linear) = 10^NF_pa/10
F_c = cable loss equivalent noise factor (linear) = 10^NF_c/10 > = 1
F_stb = set top box 1st transistor circuit noise factor (linear) = 10^NF_stb/10 > = 1
A_pa = preamp gain ratio (linear) = 10^G_pa/10 > = 1
A_c = cable loss ratio (linear) = 10^L_c/10 < = 1

It is important to note that Frii's formula is specified in linear ratio terms even though component noise figures and gains are typically expressed in dB. We will see the significance of this later. So lets start with a perfect system.

Case 1: Perfect system with unity gain pre-amp
--------------------------------------------------

Defined as

F_pa = 1 noiseless preamp (NF_pa=0dB)
F_c = 1 ideal cables (NF_c=0dB)
F_stb = 1 noiseless stb (NF_stb=0dB)
A_pa = 1 unity gain preamp (G_pa=0dB)
A_c = unity gain cables (L_c=0dB)

Substituting into (4) we get

F_sys = F_pa + (F_c-1)/A_pa + (F_stb -1)/(A_pa*A_c)
= 1 + (1-1)/1 + (1-1)/(1*1) = 1 (linear) and substituting into (2) gives

NF_sys = 10log(1) = 0dB and substituting into (3) gives

NM = 3dB - NF_sys(db) = 3dB - 0dB = 3dB (the best an ideal system could ever get assuming the specified SNR_in). So what happens when we add in a noisy STB?


Case 2: lossless cables, ideal unity gain pre-amp and noisy STB
----------------------------------------------------------------

Defined as

F_pa = 1 noiseless preamp (NF_pa=0dB)
F_c = 1 ideal cables (NF_c=0dB)
F_stb = 10^(NF_stb/10) = 4 noisy stb (NF_stb=6dB)
A_pa = 1 unity gain preamp (G_pa=0dB)
A_c = unity gain cables (L_c=0dB)

Substituting into (4) we get

F_sys = F_pa + (F_c-1)/A_pa + (F_stb -1)/(A_pa*A_c)
= 1 + (1-1)/1 + (4-1)/(1*1) = 4 (linear) and substituting into (2) gives

NF_sys = 10log(4) = 6dB and substituting into (3) gives

NM = 3dB - NF_sys(db) = 3dB - 6dB = -3dB

A noisy STB 1st circuit has eaten up all of our noise margin. What happens when we add in a quiet unity gain preamp?


Case 3: lossless cables, unity gain but quiet pre-amp and noisy STB
----------------------------------------------------------------

Defined as

F_pa = 10^(N_pa/10) = 1.6 quiet preamp (NF_pa=2dB)
F_c = 1 ideal cables (NF_c=0dB)
F_stb = 10^(NF_stb/10) = 4 noisy stb (NF_stb=6dB)
A_pa = 1 unity gain preamp (G_pa=0dB)
A_c = 1 unity gain cables (L_c=0dB)

Substituting into (4) we get

F_sys = F_pa + (F_c-1)/A_pa + (F_stb -1)/(A_pa*A_c)
= 1.6 + (1-1)/1 + (4-1)/(1*1) = 4.6 (linear) and substituting into (2) gives

NF_sys = 10log(4.6) = 6.6dB and substituting into (3) gives

NM = 3dB - NF_sys(db) = 3dB - 6.6dB = -3.6dB

So before I understood Frii's formula I would have guessed that the NM would decrease by 6dB from the STB plus another 2dB from the preamp giving a net NF_sys of 8dB. Instead it only degrades the NM by 6.6dB. Stated another way, 2dB of preamp NF only increases the system noise figure by .6dB. The reason is that gains, losses and noise figures specified in dB don't add nicely when inserted into Frii's formula. log(a+b) does not equal log(a)*log(b)!


So lets add in cable losses.

Case 4: lossy cables, unity gain but quiet pre-amp and noisy STB
----------------------------------------------------------------

Defined as

F_pa = 10^(N_pa/10) = 1.6 quiet preamp (NF_pa=2dB)
F_c = 10^(NF_c/10) = 2.5 nonideal cables (NF_c=4dB)
F_stb = 10^(NF_stb/10) = 4 noisy stb (NF_stb=6dB)
A_pa = 1 unity gain preamp (G_pa=0dB)
A_c = 10^(L_c/10)=0.3981 lossy cables (L_c=-4dB)

Substituting into (4) we get

F_sys = F_pa + (F_c-1)/A_pa + (F_stb -1)/(A_pa*A_c)
= 1.6 + (2.5-1)/1 + (4-1)/(1*0.3981) = 10.65 (linear) and substituting into (2) gives

NF_sys = 10log(10.65) = 10.3dB and substituting into (3) gives

NM = 3dB - NF_sys(db) = 3dB - 10.3dB = -7.3dB


Adding an additional -4db of cable loss increases the system noise figure by 3.7dB. Interestingly this is closer to what I might have assumed if I simply added 4db of cable loss to the NF_sys from the previous case.


So lets look at how preamp gain affects things by looking at Frii's formula in the limit as G_pa->inf.

Case 5: lossy cables, quiet pre-amp with infinite gain and noisy STB
-----------------------------------------------------------------------

Defined as


F_pa = 10^(N_pa/10) = 1.6 quiet preamp (NF_pa=2dB)
F_c = 10^(NF_c/10) = 2.5 nonideal cables (NF_c=4dB)
F_stb = 10^(NF_stb/10) = 4 noisy stb (NF_stb=6dB)
A_pa = inf preamp with infinite gain (G_pa= inf dB)
A_c = 10^(L_c/10)=0.3981 lossy cables (L_c=-4dB)

Substituting into (4) we get

F_sys = F_pa + (F_c-1)/A_pa + (F_stb -1)/(A_pa*A_c)
= 1.6 + (2.5-1)/inf + (4-1)/(inf*0.3981)
= 1.6 + 0 +0
= 1.6 (linear) and substituting into (2) gives

NF_sys = 10log(1.6) = 2dB and substituting into (3) gives

NM = 3dB - NF_sys(db) = 3dB - 1dB = 2dB

So one can see how increasing the preamp gain allows one to ignore cable losses and STB noise. Large numbers in the denominator blows associated terms away in Frii's formula.

As G_pa--> inf then NF_sys --> N_pa.


Of course, as has been noted, you are stuck with the preamp noise figure and can get no more improvement in noise margin. Clearly infinite preamp gain is not practical so one might ask what is the minimum pre-amp gain to achieve positive NM?

Case 6: lossy cables, quiet pre-amp with minimum gain and noisy STB
-----------------------------------------------------------------------

Defined as


F_pa = 10^(N_pa/10) = 1.6 quiet preamp (NF_pa=2dB)
F_c = 10^(NF_c/10) = 2.5 nonideal cables (NF_c=4dB)
F_stb = 10^(NF_stb/10) = 4 noisy stb (NF_stb=6dB)
A_pa = to be solved for
A_c = 10^(L_c/10)=0.3981 lossy cables (L_c=-4dB)

To determine the minimum preamp gain set NM=0dB in (3) and solve to get

NF_sys (max) = 3dB and subsequently F_sys (max) = 10^(NF_sys/10) = 2


solving (4) for A_pa and substituting in F_sys (max) gives


A_pa_min = (F_c-1+(F_stb-1)/A_c)/(F_sys-F_pa)
= (2.5 - 1 + (4-1)/.3981)/(2-1.6)
= 22.6

Converting to dB gives the minimum preamp gain to reach the threshold of positive noise margin as

G_pa_min = 10log(A_pa) = 13.5 dB

So 13.5dB of preamp gain gets you back to 0db NM but you need an infinite amount of additional gain to get the last possible 1dB of remaining NM back.

So one question that still remains is what is the sweet spot selecting a preamp gain given the outlined diminishing returns. I suspect you could pick the maximum preamp gain hat would not overload the preamp or STB 1st circuit but I am not sure how one would determine that.

Another question is how can Frii's formula be used to analytically determine the preamp gain that would cause it to be dominant? I believe GroundUrMast outlined in his response to me something in terms of baseline receiver noise, baseline antenna noise and gain but I haven't been able to frame it in terms of Frii's formula.


Frii's formula for the above 3 stage system, explicitly written in terms of inputs expressed in dB's, is quite nonlinear and given as:

NF_sys = 10log[10^NF_pa/10 + (10^NF_c/10 -1)/10^G_pa/10 + (10^NF_stb/10 -1)/10^(G_pa+L_c)/10]

there is probably some way to rearrange the above equation so that it is clear how much bigger G_pa must be over L_c and NF_stb in order for everything down stream of the preamp to be ignored but I think I'll stop there for now.

Last edited by mmbridges; 4-Dec-2013 at 2:02 AM.
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Old 4-Dec-2013, 3:00 AM   #7
GroundUrMast
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If it's not obvious to everyone at this point... My math skills are limited to the level of a graduate of senior level public high school algebra circa the mid 70's.

I admire @mmbridges apparent skills and have to presume that those skills have been honed by significantly more education than mine.

Not to discourage or minimize the value of accurate evaluation... Such skill and ability is vital in applications such as high reliability terrestrial microwave or space communication links. However, given that OTA grade amplifiers are not provided with the necessary precision specifications, it seems to me that I'm stuck with what are relatively 'crude' estimations of expected performance.

@mmbridges, welcome, and thank you for your contribution(s) to the forum.
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Old 6-Dec-2013, 1:55 AM   #8
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Hi GroundUrMast,

It was not obvious to me. However, I have seen the quality your posts and what is obvious to me is your superior knowledge and expertise in the subject matter.

I am new to this subject and am learning a lot hence my going down the Frii's formula Noise figure analysis path. Looks like this topic has been covered quite extensively here and on other sites. I will add one additional view of things but fist another

-------------------------------------------------
DISCLAIMER
1) The following analysis and comparisons are theoretical only and based upon published preamp specs.
2) They do not include multipath or other signal quality effects
3) I understand that for most setups, small improvements in Noise Margin (NM) may end up being irrelevant but for fringe area situations like mine I need to squeeze out every bit of NM improvement I can.
---------------------------------------------------

Since equations tend to make peoples eyes glaze over I decided to create a plot that may provide additional insight. first one more equation:

Let the improvement in Noise Margin obtained from adding a preamp over no using one be expressed as

delta_NM_pa (dB) = 10log(F_sys_no_pa/F_sys_w_pa)

where

F_sys_no_pa = 1+(F_c-1)+(F_stb-1)/A_c

and

F_sys_w_pa=F_pa+(F_c-1)./A_pa+(F_stb-1)./(A_pa*A_c)

and all remaining variables have been previously defined.


delta_NM_pa is a figure of merit that will allow one to see preamp improvements in NM without having to talk about what the absolute NM of the system is. If the absolute NM without a preamp is positive and much greater than delta_NM_pa (i.e. NM_no_pa >> delta_NM_pa and NM_no_pa >0) then the following discussion is moot as a preamp buys you little.

If the absolute NM without a preamp is negative and much smaller than delta_NM_pa (i.e. NM_no_pa << delta_NM_pa and NM_no_pa < 0) then the following discussion is also moot as there is nothing a preamp can do to get you back to having positive NM.

However, if absolute NM without a preamp is 0dB, you are on the edge and a preamp can possibly help. What follows is a discussion of this situation.


Because there are many variables in the delta_NM_pa equation, in order to generate a plot a few variables had to be set. I set the following variables to meet my particular setup:

Single TV, no splitters, 50ft of RG6 one, a Zenith DTT-901 set top box, insertion losses from one coax grounding block and one house main antenna wall plate.

This results in the following two fixed parameters:

NF_stb = 6dB for set top box NF
L_c=-4db for lumped cable and insertion losses

Drum roll...... Here it is:



The above is a contour plot showing delta_NM_pa as a function of preamp NF and preamp gain. contour lines represent levels of constant NM improvement. On this plot I have also placed 7 preamps. A 3D representation of the contour plot would look like a hill with the highest point in the upper left hand corner.

So to perform a sanity check to see if this plot makes sense, lets consider a fictitious preamp with NF_pa=4dB and variable gain that can be adjusted between 0 and infinity. We start with the preamp gain set to G_pa=0dB which is on the x-axis at 4. As we start increasing the preamp gain we move vertically on the plot crossing constant contour lines of increasing NM improvement as we go. What we see is that as the gain is goes to infinity we approach but can not cross the 6dB NM improvement contour. This makes sense because in the limit, the system noise figure without the preamp is

NF_sys_no_pa=set top box NF + cable losses = 6dB + 4dB = 10dB

furthermore the system noise figure with the infinite gain preamp is

NF_sys_w_pa = NF_pa = 4dB.

Consequently the max possible NM improvement for this preamp on this system will be NF_sys_no_pa-NF_sys_w_pa = 10-4 = 6dB.


If we perform a similar test on a preamp with NF_pa=0dB and increase the gain we will find that we approach a constant NM improvement contour of 10dB. This is theoretically the best we could ever do on this system. Similarly if we choose a particularly noisy preamp with NF_pa =1 0dB and crank up the gain, we will never be able to cross the 0dB constant NM improvement contour. As such one will never be able to get a positive noise margin improvement and are better off not using this preamp at all.


Now one other thing about rules of thumb for selecting preamp gain and estimating NM improvement. One quick technique (i.e. one that doesn't involve Frii's formula) to estimate NM improvement from preamp, is to use the infinite gain case (i.e. delta_NM_pa = cable losses + NF_stb - NF_pa). This works very well if you use GroundUrMast's rule of thumb of setting preamp gain to be 5-10dB above the distribution losses(i.e. cable+splitter losses+tuner NF).

Let's see by examining a preamp with NF_pa=1dB. The quick "rule of thumb" based estimate of NM improvement is 10-1=9dB. If we set the preamp gain using the upper end of GroundUrMast's rule of thumb then G_pa=10+10=20dB. To find the actual NM improvement we look at the (1,20) point on the plot an see it would lie on about the 8.7dB contour. This is only 0.3dB off! However, say you simply set the gain equal to the distribution losses (i.e. G_pa=10dB). You would then be on a 6.7dB contour and more like 3.3dB off on the NM improvement estimate. The extra 5-10dB that GroundUrMast adds to the distribution losses when choosing a pre-amp gain is very important.

In general you want your preamp gain to put you in the portion of the contour plot where the contour lines are close to being vertical. This is the equivalent of having the preamp gain being "sufficiently large" to ignore the downstream losses.


Now looking at the various preamps in terms of NM improvement, the Kitztech 200 is the best for my particular setup since I am in a fringe area and have no chance of overloading the preamp. However it is important to note that the top 4 are all within 2 db of each other. The KT200 puts me just 0.5dB away from the maximum theoretical NM improvement which is quite nice if you are like me, and trying to squeeze out every bit of noise margin improvement you can. (Note: I mistakenly plotted the KT200 at 26dB instead of 24dB however when re-ran the calc the deltaNM improvement was still about 9.5dB).

As far as the generality of above contour plot is concerned, it mus be reemphasized that it was produced using a specific set of cable losses and set top box noise figure. Although different systems will have different plots, they will in general have the same shaped contours. The upper left and portion of the plot will have a higher hill top for higher distribution losses.

Last edited by mmbridges; 7-Dec-2013 at 2:27 PM. Reason: spelling and KT200 plot error disclosure
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