This post discuss about the thermal noise in RC low pass filter. Using the noise equivalent model using resistor with a voltage source, which gets passed through a no noise RC low pass filter. The noise power at the output is computed by integrating the output voltage spectral density over all frequencies.

Consider a simple RC low pass filter with resistor  and capacitance  as shown in the figure below.

Figure : Noise equivalent model of RC low pass filter (Reference : Figure 9.3-6 in Communication Systems – An introduction to Signals and noise in Electrical Communication by A. Bruce Carlson, Paul Crilly, Janet Rutledge)

As discussed in the post on thermal noise and AWGN, the resistor  is generating a noise having power  having power spectral density

.

The noise voltage gets applied to a noise less RC low pass filter having transfer function,

, where

 is the -3dB bandwidth of the filter.

The power spectral density of the output signal is,

.

Integrating the output power spectral density over all frequencies from   will give the noise voltage at the output, i.e.

.

Computing  by plugging in the expression for ,

.

(Note: The neat trick behind integration described below)

Plugging in  and , the noise power at the output is,

.

Summarizing, the mean square voltage at the output of the RC low pass filter is,

.

Plugging in capacitance values from 1pF to 1nF, the root mean square voltage is shown below.

Observations:

a) As expected from the equation, when the capacitance is increased, the noise voltage reduces (as the filter bandwidth is getting narrower).

b) The noise power at the output is independent of resistor . Increasing the resistance will increase the noise power , but decreases the filter bandwidth . Similarly decreasing the resistor will reduce the noise power, but increases the filter bandwidth. These two effects cancel each other leaving the noise power dependent only on the temperature  and capacitance .

References:

a) Communication Systems – An introduction to Signals and noise in Electrical Communication by A. Bruce Carlson, Paul Crilly, Janet Rutledge

b) List of Trigonometric Identities from wikipedia

c) Differentiation rules from wikipedia

d) Johnson-Nyquist Noise from wikipedia

Note :

Liked the neat trick using  substitution for solving the integration over all frequencies in a first order RC low pass filter. Did some googling to find that out…and the steps are as follows.

Let , then   and the integration limits are

, .

Also, note that (thanks to the List of Trigonometric Identities from wikipedia, Differentiation rules from wikipedia)

,

Plugging all the above,

. Neat!

Related posts:

1. Digital implementation of RC low pass filter
2. Noise Figure of resistor network
3. Thermal Noise and AWGN

Following the discussion on thermal noise and it’s modeling and noise figure computation for a simple resistor network, in this article let us discuss the Noise Figure of cascaded stages.

Definition of Noise Figure

From Section 2.3.2 of RF Microelectronics, Behzad Razavi, the noise figure is defined as

, where

 is the input signal to noise ratio and

 is the output signal to noise ratio.

Consider that the system has a gain  and is the noise added by the system.

With that,

The Noise Figure is,

.

From the previous post on thermal noise, we know that noise power over bandwidth at temperature Kelvin is,

.

With that the noise figure equation can be modified as,

.

Two key equations to remember: 

a) Noise added by the noise sources in system is,

b) Noise at the output of the system is,

Example

For example, consider a device of bandwidth and having a gain  with noise figure of . What is the noise power seen at the output of the device?

Solution: Assuming temperature of 300K, the noise seen at the input of the device

.

.

Expressing in deciBels, the noise power seen at the output of the device is,

.

Noise Figure of cascade of two stages

Consider a system with cascade of two gain stages having gains of , and with noise figure of and respectively.

Figure : Cascaded gain stages with noise power output

The noise power at the output of the first stage is the sum of

a) Noise power at the input terminal of first gain stage  which is amplified by gain .

b) Noise power generated by the first gain stage 

Summing them up, the noise seen at the output of first gain stage is,

* Note : The sum of powers is possible because it is assumed that the input noise and the noise added by the system are uncorrelated i.e.

, where

when  and are uncorrelated.

Related posts:

1. Noise Figure of resistor network
2. Example of Cascaded Integrator Comb filter in Matlab
3. Thermal noise of RC low pass filter

The post on thermal noise described the noise produced by resistor  ohms over bandwidth  at temperature Kelvin. In this post, let us define the noise voltage at the input and output of a resistor network and further use it to define the Noise Figure of such a network.

From Section 2.3.2 of RF Microelectronics, Behzad Razavi, the noise figure is defined as

, where

is the input signal to noise ratio and

is the output signal to noise ratio.

Note : Some textbooks refer to this term as Noise Factor.

Noise Figure of a resistor network

Consider a simple resistor circuit with  as the input voltage,  source resistance and  as the parallel resistance (shown below) :

Figure : Resistor network (Reference : Figure 2.30(a) of  RF Microelectronics, Behzad Razavi)

The noise voltage due to the source resistance  is

.

The Signal to Noise Ratio at the input is,

.

The signal voltage at the output is,

The noise voltage at the output is,

Figure : Equivalent resistor noise model for two parallel resistor (Reference : Figure 2.30(b) of  RF Microelectronics, Behzad Razavi)

.

The Signal to Noise Ratio at the output is,

.

Calculating,

.

The Noise Figure is,

Observations

a) Per the Maximum Power Transfer theorem the source and load resistance should be equal i.e . However that condition does not coincide with the minimum noise figure. The source and load resistance and equal results in  (3dB in decibels)

b) Noise Figure is minimized by maximizing resistance  OR minimizing source resistance .

Reference

RF Microelectronics, Behzad Razavi

Related posts:

1. Noise Figure of cascaded stages
2. Thermal noise of RC low pass filter
3. Thermal Noise and AWGN

SOLVED the Rubik’s cube !!!  

After 6 months, 2 cube’s and countless twists and turns, extremely glad to reach here.

Will enjoy the beauty of the solved cube for couple of days before breaking it and going over the whole journey again….

(Thanks dear Kunju for introducing me to the cube)

Disclosure :

After solving the first two layers, going got really tough. Couple of pointers from Rubik’s original website helped me proceed.

Related posts:

1. Polyphase filters for interpolation
2. Summary – feedback on [dspLog], July 2008

A friend called me up couple of days back with the question – How white is AWGN? I gave him an answer over phone, which he was not too happy about. That got me thinking bit more on the topic and the result is this post – brief write up on thermal noise and it’s modelling as Additive White Gaussian Noise aka AWGN.

Thermal Noise

Quoting from Chapter 9.3 of Communication Systems – An introduction to Signals and noise in Electrical Communication by A. Bruce Carlson, Paul Crilly, Janet Rutledge

“Thermal Noise is the noise produced by the random motion of charged particles (usually electrons) in conducting media.”

With a resistor  ohms at temperature Kelvins, the noise voltage due to random electron process is a Gaussian distributed (thanks to Central Limit Theorem) variable with zero mean, and variance of

, where

is the Boltzmann constant Joules/Kelvin,

is the Planck constant Joules second.

The spectral density of the thermal noise is shown as,

.

Assuming   and using the Taylor series approximation, the equation simplifies to :

.

Note : 

a) Mr. Clay S Turner in his write up “Johnson-Nyquist Noise”, Turner C. S., Jan 2007 has derived the above expression – I did not understand the derivation – one of my many TO-DO’s.  

b) The fundamental paper’s discussing this topic is (thanks to wiki entry on Johnson-Nyquist noise)

 “Thermal Agitation of Electric Charge in Conductors”, (1928) H. Nyquist,

 “Thermal Agitation of Electricity in Conductors”, (1928) J. Johnson,

Have not gone through both. 

c) The Taylor Series results used for the approximation

From the plot of the spectral density of thermal noise over frequency, can see that the noise is flat frequency spectrum till around 100GHz or so and starts to fall off at around 1TeraHz.

 Figure : Spectral density of thermal noise

Given the above plot, for most of the practical systems which we are dealing with today (i.e. with carrier frequency less than 10GHz or so), can assume that the noise spectral density is flat, i.e.

.

Thus for a bandwidth , the root mean square noise voltage is,

The mean square voltage is,

Note :

The factor  is because the noise is real with a symmetric frequency spectrum. Multiplication by 2 suffice to consider both the positive and negative frequencies.

Matlab/Octave script to get the above plot

clear all; close all;
k = 1.38e-23; % Boltzmann constant
h = 6.62e-34; % Planck's constant
T_kelvin = 300; % temperature in Kelvin
f_hz 	 = logspace(0,14,10000);
R_ohm    = 50;
Gf 	   = 2*R_ohm*h*f_hz./(exp((h*f_hz)/(k*T_kelvin)) - 1);
Gf_approx1 = 2*R_ohm*k*T_kelvin*(1-h*f_hz/(2*k*T_kelvin));

semilogx(f_hz,Gf,'bs-');
hold on; grid on;
semilogx(f_hz,Gf_approx1,'r-');
axis([1 10^14 10^-21 0.5*10^-18]);
legend('exact', 'approx');
xlabel('frequency, Hz');
ylabel('voltage spectral density, V^2/Hz');
title('Spectral density of thermal noise');

Plugging in some numbers :

For  ohms,

Temperature (26.85 deg Celsius),

Bandwidth ,

will result in rms noise voltage of

1 nano Volts

(Thanks to wiki entry on Johnson Nyquist Noise for this)

Thevenin Equivalent Model

Thevenin equivalent model of the thermal resistance noise is shown below, where the it is modeled as a voltage source in series with a noiseless resistor.

Figure : Thevenin equivalent model with matched load (Figure 9.3-3 in Communication Systems – An introduction to Signals and noise in Electrical Communication by A. Bruce Carlson, Paul Crilly, Janet Rutledge)

Assuming that the load resistance is  (per maximum power transfer theorem), the rms voltage seen across the load resistance is,

The power is,

.

Note :

a) The noise power is independent of the resistance .

b) Above is the single sided noise power

Noise Power in dBm

Typically the noise power is expressed in dBm (0dBm = 1 milli Watts). The noise power in dBm for 1Hz at room temperature

In general, for a bandwidth , the noise power is

.

Modeling white noise

Typically, we can write spectral density of white noise as

.

Further, from Weiner – Khinchin theorem, it is known that power spectral density and autocorrelation are Fourier transform pairs, i.e. the autocorrelation is,

.

Figure : Power spectral density and autocorrelation of white noise

Note: The term 1/2 is to indicate that power is symmetric across both positive and negative frequencies.

In the simulation models used in the blog, the randn() function provided by Mathworks is used to generate White Gaussian Noise. Using a small Matlab code snippet, can see that over many realizations the simulated power spectral density tends to be white (i.e. constant across frequency).

clear all;
n_fft = 2048; n_vec = 1000;
nt  = randn(n_vec,n_fft);
ntf = 1/sqrt(n_fft)*fft(nt,[],2);
ntf_pwr     = ntf.*conj(ntf);
ntf_pwr_avg = mean(ntf_pwr);
plot([-n_fft/2:n_fft/2-1]/n_fft,10*log10(fftshift(ntf_pwr_avg)));
axis([-0.5 0.5 -5 5]); grid on;
ylabel('power spectral density, dB/Hz');
xlabel('normalized frequency');
title('power spectral density of white noise');

Figure : Simulated spectrum of white noise

Further, can see an impulse for the simulated auto-correlation of the noise vector generated from randn().

clear all;
n_len = 10^4;
nt = randn(1,n_len);
n_ac = 1/n_len*conv(fliplr(nt),nt);
plot([-n_len+1:n_len-1],n_ac);
xlabel('delay, tau');
ylabel('autocorrelation');
title('autocorrelation of white noise');

Figure : Simulated autocorrelation of white noise

References

Communication Systems – An introduction to Signals and noise in Electrical Communication by A. Bruce Carlson, Paul Crilly, Janet Rutledge

Johnson-Nyquist Noise”, Turner C. S., Jan 2007

Wiki entry on Weiner – Khinchin theorem

Wiki entry on Johnson Noise 

randn() function by Mathworks

Related posts:

1. Thermal noise of RC low pass filter
2. Noise Figure of resistor network
3. Noise Figure of cascaded stages

An earlier post we discussed hard decision decoding for a Hamming (7,4) code and simulated the the bit error rate. In this post, let us focus on the soft decision decoding for the Hamming (7,4) code, and quantify the bounds in the performance gain.

Hamming (7,4) codes

With a  Hamming code, we have 4 information bits and we need to add 3 parity bits to form the 7 coded bits.

The coding operation can be denoted in matrix algebra as follows:

where,

 is the message sequence of dimension ,

 is the coding matrix of dimension ,

 is the coded sequence of dimension .

Using the example provided in chapter eight (example 8.1-1) of Digital Communications by John Proakis , let the coding matrix  be,

.

This matrix can be thought of as,

 ,

where,

 is a  identity matrix and

 is a  the parity check matrix.

Since  an identity matrix, the first  coded bits are identical to source message bits and the remaining  bits form the parity check matrix.

This type of code matrix  where the raw message bits are send as is is called systematic code.

Assuming that the message sequence is , then the coded output sequence is :

, where

,

,

.

The operator  denotes exclusive-OR (XOR) operator.

 The matrix of valid coded sequence  of dimension 

Table: Coded output sequence for all possible input sequence

Minimum distance

Hamming distance computes the number of differing positions when comparing two code words. For the coded output sequence listed in the table above, we can see that the minimum separation between a pair of code words  is 3.

If an error of weight  occurs, it is possible to transform one code word to another valid code word and the error cannot be detected. So, the number of errors which can be detected is .

To determine the error correction capability, let us visualize that we can have  valid code words from possible  values. If each code word is visualized as a sphere of radius , then the largest value of  which does not result in overlap between the sphere is,

where,

 is the the largest integer in .

Any code word that lies with in the sphere is decoded into the valid code word at the center of the sphere.

So the error correction capability of code with  distance is .

In our example, as , we can correct up-to 1 error.

Soft decision decoding

Let the received code word be,

, where

and

is the transmit code word,

is the additive white Gaussian noise with mean and variance  and

 form the elements of the code word.

Given that there are known code words, the goal is to find correlation of received vector with each of the valid code words.

The correlation vector is,

where,

 is the received coded sequence of dimension ,

 is the matrix of valid code words sequence of dimension  and

 is the vector of correlation value for each valid code word and is of dimension

.

From the correlation values, the index of the location where  is maximized corresponds to the maximum likelihood transmit code word.

Note:

The term is to given weights for the code words i.e.0 is given a weight -1, 1 is given a weight 1.

Hard decision decoding

To recap the discussion from the previous post, the hard decision decoding is done using parity check matrix .

Let the system model be,

, where

 is the received code word of dimension ,

 is the raw message bits of dimension ,

 is the raw message bits ,

 is the error locations of dimension .

Multiplying the received code word with the parity check matrix,
.

The term  is called the syndrome of the error pattern and is of dimension .  As the term , the syndrome is affected only by the error sequence.

Assuming that the error hits only one bit,

a) There are can be  possible error locations.

b) If the syndrome is 0, then it means that there is no errors.

c) The value of syndrome takes one among the valid 7 non-zero values. From the value of syndrome we can figure out which bit in the coded sequence is in error and correct it.

Note :

a) If we have more than one error location, then also the syndrome will fall into one of the 8 valid syndrome sequence and hence cannot be corrected.

b) The chosen Hamming (7,4) coding matrix , the dual code is,

.

It can be seen that modulo-2 multiplication of the coding matrix  with the transpose of the dual code matrix  is all zeros i.e

.

Asymptotic Coding gains

From Chapter 12.2.1 and Chapter 12.2.1 of Digital Communication: Third Edition, by John R. Barry, Edward A. Lee, David G. Messerschmitt, the asymptotic coding gain with soft decision decoding and hard decision decoding is given as,

.

where,

is the coding rate,

is the minimum distance between the code words and

is the maximum number of errors which can be corrected.

Simulation Model

The Matlab/Octave script performs the following

(a) Generate random binary sequence of 0′s and 1′s.

(b) Group them into four bits, add three parity bits and convert them to 7 coded bits using Hamming (7,4) systematic code

(c) Add White Gaussian Noise

(d) Perform hard decision decoding – compute the error syndrome for groups of 7 bits, correct the single bit errors

(e) Perform soft decision decoding

(f) Count the number of errors for both hard decision and soft decision decoding

(g) Repeat for multiple values of  and plot the simulation results.

Click here to download Matlab/Octave script for computing BER for BPSK in Hamming (7,4) code with soft and hard decision decoding.

Figure : BER plot for Hamming (7,4) code with soft and hard decision decoding

Observations

a) At bit error rate close to , can see that the coding gains corresponding to hard and soft decision decoding is tending towards the asymptotic coding gain numbers.

References

Digital Communications by John Proakis

Digital Communication: Third Edition, by John R. Barry, Edward A. Lee, David G. Messerschmitt

Related posts:

1. Hamming (7,4) code with hard decision decoding
2. Soft Input Viterbi decoder
3. Viterbi with finite survivor state memory

My friend and colleague Mr. Vineet Srivastava pointed me to a nice article on  clock jitter - Clock Jitter Effects on Sampling : A tutorial – by Carlos Azeredo-Leme, IEEE Circuits and Systems Magazine, Third Quarter 2011. In this post, let us discuss the total Signal to Noise Ratio at the output of an analog to digital converter (ADC) accounting for errors due to sampling clock jitter and quantization noise.

Clock jitter

The error in the sampling clock results in an error in the sampled voltage as show in the figure below. Can see that the jitter in the clock  results in an error voltage .

Figure : Voltage error caused by jitter

For a  signal ,

the ratio of  the voltage error and the time delta is the slope i.e.

.

Re-ordering, and replacing with the error voltage ,

Taking the mean square error,

where

is the variance of the clock jitter.

The signal to noise ratio at the output of ADC due to jitter alone is,

.

Assuming that the signal is sinusoidal, i.e.

.

The first derivative is,

The signal power is,

The error power is,

The signal to noise ratio is,

.

Expressing in decibels,

Quantization Noise

An input signal with swing from of  volts will get discretized by an bit ADC to levels.

Figure : Input and output from an N bit ADC (with N=3)

The maximum error is .

For a linearly varying input as show above, the error voltage is having a uniform distribution

.

The error voltage is a periodic repetition of the sawtooth waveform,

The power of the error signal is,

.

For an input sinusoidal signal ,the signal power is,

.

The signal to quantization noise ratio is,

Expressing in decibels,

.

Total Signal to Noise Ratio

The overall signal to noise ratio for an ADC with both quantization noise and jitter is,

The simple Matlab script computes the total SNR for 10bit ADC with varying input frequency for different rms jitter specifications.

clear all; close all;
fs_MHz 	= 2000;  % sampling clock
fm_MHz 	= [0.1:1:1000]; % signal frequency
t_rms_ps= [5 10 20 50 100]; % rms jitter
tn 	= [0:1/fs_MHz:10];
nBit	= 10; % number of bits of the ADC
snr_dB  = zeros(length(t_rms_ps),length(fm_MHz));
for (jj = 1:length(t_rms_ps))
	tn_jitter	= tn + t_rms_ps(jj)*1e-6*randn(size(tn));
	for (ii = 1:length(fm_MHz))
		xt = (exp(j*2*pi*fm_MHz(ii)*tn));
		yt = (exp(j*2*pi*fm_MHz(ii)*tn_jitter));
		yt = floor(yt*2^(nBit))./2^(nBit);
		err = yt - xt;
		sig_pwr = xt*xt'/length(xt);
		err_pwr = err*err'/length(err);
		snr_dB(jj,ii) = 10*log10(sig_pwr/err_pwr);
	end
end
semilogx(fm_MHz,snr_dB.');
grid on;axis([0.1 1000 20 70]);
xlabel('freq MHz'); ylabel('SNR, dB');
legend('5ps','10ps','20ps','50ps','100ps');
title('SNR for 10bit ADC with different RMS jitter values');

Figure : Total SNR  for a 10bit ADC with different RMS jitter specifications (simulation of the Figure 3 in Clock Jitter Effects on Sampling : A tutorial)

Observations 

a) For lower frequency signals, quantization error dominates.

b) For higher frequency signals, the jitter error dominates.

References

Clock Jitter Effects on Sampling : A tutorial – by Carlos Azeredo-Leme, IEEE Circuits and Systems Magazine, Third Quarter 2011

Aperture Uncertainty and ADC System Performance by Brad Brannon and Allen Barlo, Analog Devices Application Note AN501

Little Known Characteristics of Phase Noise by Paul Smith, Analog Devices Application Note, AN-741

Taking the Mystery out of the Infamous Formula,  ”SNR = 6.02N + 1.76dB,” and Why You Should Care  by Walt Keste, Analog Devices Tutorial MT-001

Related posts:

1. Signal to quantization noise in quantized sinusoidal
2. Noise Figure of resistor network
3. Solved objective questions (GATE)