My friend Mr. Balaji volunteers for Vibha, a non-profit  organization whose mission is to ensure that every underprivileged child attains his or her right to education, health and opportunity.

Vibha, which was founded in 1991 has a volunteer network of 825 members spread across Atlanta, Austin, Bay Area, Boston, Chicago, Dallas, Houston, Jacksonville, Los Angeles, Milwaukee, Twin Cities – Minnesota, New York, Philadelphia, Sacramento, Washington DC and several other cities across the US and India. Till date, we have supported about 190 projects in India and the US. You may read more about their projects here.

To support Vibha’s cause, Mr Balaji has set up a donation page – The Dream Mile … A few miles for a million dreams . The Dream Mile 5k/10k Run/Walk is Vibha’s flagship event for increasing awareness and raising funds to help underprivileged children. Vibha has chosen  My New Red Shoes as one of the beneficiaries of this event.

To find out more about My New Red Shoes, please watch the video below.

To find out more about Alamb, a project by Vibha in India, please watch the video below.

If you like some of the work done by Vibha, please consider making a donation to the cause. As Mr. Balaji has mentioned:

$120 can provide education to a child for a whole year

$50 can facilitate the rehabilitation of a mentally handicapped child for a year

$40 can sponsor the non-formal education of a child for a whole year.

To donate for this cause, please visit the donation page The Dream Mile … A few miles for a million dreams

Thanks again for your support.

Related posts:

1. OT: Prof. Randy Pausch’s lecture in Oprah show
2. ICCBN 2008, July 17-20 2008, IISc, Bangalore
3. MCDES 2008 at Indian Institute of Science, Bangalore

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

Fundamentals of Wireless Communication, David Tse, Pramod Viswanath

Digital Communications by John Proakis

Digital Communications: Fundamentals and Applications by Bernard Sklar

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

Digital Signal Processing by John G. Proakis, Dimitris K Manolakis

Discrete-Time Signal Processing – by Alan V. Oppenheim, Ronald W. Schafer, John R. Buck

Note: Images, thanks to Amazon.com

Related posts:

1. Download free e-book on error probability in AWGN
2. OCW: Communication System Design
3. Negative Frequency

We have installed Google FriendConnect on dspLog.com. With Google Friend Connect, you can:

(a) You can interact with other members who have similiar interests. You will come to know the list of other sites (apart from dspLog.com) where the members have joined. You can add a member as a friend and so on.

(b) You can invite friends from orkut, Google Talk and other social networks and contact lists to join.

(c) You need not create a new account. You can sign in with your Google, Yahoo, AIM or OpenID account.

If you wish to join dspLog community via FriendConnect, please click ‘Join this Site‘ button on the sidebar.

From Google support page on FriendConnect

You can become friends with any member of a Friend Connect site, regardless of what social networks they belong to and what account they used to sign up. Friends you make on a Friend Connect site will be friends of yours on any other sites you both belong to.

Regarding privay

Google Friend Connect never provides site owners with your private sign-in information. Google Friend Connect and the service you use to sign in — not site owners — validate your sign-in credentials.

Google stores and shares as little information as possible to be able to provide the best possible experience.

If you wish to join dspLog community via FriendConnect, please click ‘Join this Site‘ button on the sidebar.

For more details on Google FriendConnect, please click here to visit Google’s support page.

Related posts:

1. Migration to Who’s Who theme
2. Support Vibha’s Dream Mile event
3. dspLog turns two! Happy Birthday!

In signal processing blocks like power estimation used in digital communication, it may be required to represent the estimate in log scale. This post explains a simple linear to log conversion scheme proposed in the DSP Guru column on DSP Trick: Quick-and-Dirty Logarithms. The scheme makes implementation of a linear to log conversion simple and small in a digital hardware like FPGA.

Consider an integer . The floating point representation is,

where,

is the exponent and

is the mantissa.

Assume that is normalized, i.e .

Taking logarithm to the base 2,

.

In digital hardware implementations, finding the exponent is simple. Its just noting the index of the first bit which is 1 starting from MSB side.

For example consider an input number .

Expressed in binary on 8 bit bus, .

The value of in this example is 3.

Now, the part which remains to be computed is the mantissa . In this example,

.

Given that lies in the range . this can be computed using a Look Up Table. The LUT can store values of input between 1 to 2. The precision requirement determines the number of elements in the LUT. Let us assume that we want to have a precision of , where . The look up table values will be as follows:

Table: Look up table values for logarithm computation

From the above look up table, we can see that mantissa of corresponds to index of . It is inituitive to note that the array index can be found out by the simple formula,

. To handle cases where this number can can be a fraction, the result is floored to the nearest integer, i.e.

So the value of in base is,

.

Once we have the number in base, conversion to any other base is simple.

.

So the number in base is,

.

Simulation Model

Simple Matlab/Octave script for computing the logarithm via the LUT based approach is provided. Click here to download Matlab/Octave script for performing linear to log conversion using LUT based approach

Figure: Linear to log conversion using LUT

Reference

DSP Trick: Quick-and-Dirty Logarithms – Ray Andraka, June 2000

Related posts:

1. Gray code to Binary conversion for PSK and PAM
2. Binary to Gray code conversion for PSK and PAM
3. Binary to Gray code for 16QAM

Last week, I received an email from Mr. Kishore. He was wondering about the physical significance of negative frequency. Does negative frequency really exist?

Though I have seen conflicting views on the net (thread in complextoreal.com, thread in comp.dsp), my perspective is that negative frequency exist. The concept of negative frequency helps me a lot to understand single sideband modulation (SSB), OFDM systems, I Q modulators etc (to name a few).

Simple explanation for negative frequency

The wiki entry on negative frequency provides a simple explanation using as an example.

We know that . This means the sign of cannot be un-ambiguously found out from observing alone. This implies that it is reasonable to think that has frequency components at both and .

Similarly, this ambiguity exists for too.

Negative frequency using Taylor’s series expansion

Thanks to the nice paper by Mr. Richard Lyons, Quadrature Signals: Complex but not Complicated, Richard Lyons.

Let us first define the magic number and so on.

The Taylor series expansion of , and are as follows.

Let us now define the Taylor series expansion of .

.

Similarly,

.

Combining the above two equations, one may write

.

If we apply , then we get,

and

.

This forms the proof that a real sinusoidal having frequency is comprised of a complex sinusoidal having a positive frequency at and a negative frequency at .

Simple Matlab example for Negative frequency

Click here to download Matlab/Octave code for plotting the spectrum of real and complex sinusoidal

Figure: Spectrum plot showing positive and negative frequency

As discussed above, the real sinusoidal has frequency components at +5MHz and -5MHz where as the complex sinusoidal has frequency component only at +5MHz.

Negative frequency in OFDM

Some of you might be familiar with the IEEE 802.11a specification where subcarriers from [-26 to -1 and [+1 to 26] are used. The subcarriers -26 to -1 corresponds to usage of negative frequency and lets try to understand it.

The equation for an OFDM transmission is,

,

where

(a) correspond to the frequency of the sinusoidal and

(b) is a rectangular window over

(c) is the symbol period.

(d) each information signal is modulated on to a complex sinusoidal having frequency of .

(e) Sum of all such modulated sinusoidals are added and the resultant signal is sent out as .

In the IEEE 802.11a specification, symbol duration is 3.2, sampling frequency is 20MHz and .

The frequencies used for modulating the ’s are , , , and so on till . Expressing in Hz, this corresponds to frequencies from 0Hz, 312.5kHz, 625kHz, 937.5kHz, 1.25MHz,…, 10MHz, 10.3125MHz,… till 19.6875MHz.

From our understanding of sampling theory, we know that with a sampling frequency of , we can only see frequencies from to .

Note: The frequency is called the Nyquist frequency.

So in our 802.11a example, what will happen to frequencies which are modulated on subcarriers lying from 10MHz till 19.6875MHz?

Quick answer: They get folded!

The frequencies from 10MHz till 19.6875MHz gets folded and seems as if they are lying from -10MHz to -312.5kHz.

Figure: Spectrum folding to negative frequency in IEEE802.11a specification

Further, folding of the spectrum to the negative frequency region does not cause any problems. Reason: ’s which where modulated on complex sinusoidals having frequencies from 0 till 10MHz did not have any negative frequency component.

Note:

In general, one can say if a sinusoidal is of frequency is sampled with a frequency and if , the frequency gets folded to a frequency within , where is an integer.

The concept of folding is well explained in Chapter 1.4.1 of [DSP: PROAKIS]. A simple example of folding which we may see in our day to day life is with a ceiling fan. One may see that the blades of the fan are rotating at a slower speed in a direction opposite to the actual rotation of the fan blades. Needless to say that the sampling frequency of our eye’s are not good enough.

Happy learning.

Reference

[DSP: PROAKIS]: Digital Signal Processing, John G. Proakis

Quadrature Signals: Complex but not Complicated, Richard Lyons

Related posts:

1. Interpreting the output of fft() operation in Matlab
2. Solved objective questions (GATE)
3. Harmonic distortion in digital sinusoidal generators

On July30th, 2008 I had sent a request for feedback to 93 subscribers who have opted to receive articles over email. As on 3rd August, I received the response from around 8 persons. Not bad, around 8.5% response. Thanks a lot for the feedback. I will summarize the response from the group and note down the action items on me.

Best in [dspLog]

The majority of the people (6) found the theoretical description to be the most useful and around 3 people found the Matlab/Octave code to be the most useful. A comment came from Mr. Eddie Maalouf who suggested that we should start encourage user participation where other engineers can help to solve technical problems etc.

Worst in [dspLog]

Two persons did not like the template and two persons felt that the theoretical description could have been made more simpler. There was also a comment that we are repeating the theoretical aspects and not going into solving problems.

Suggested topics to be covered in [dspLog]

Two persons suggested that more posts on receiver synchronization needs to be added. Two were of the opinion that more posts on OFDM and channel coding needs to be present. In general, most people commented that they wish to see more articles on the topics like MIMO, CDMA, IEEE 802.11a, 802.16, Bluetooth etc.

Frequency of posting in [dspLog]

Around 5 people were suggesting the once a week and around 2 responders were happy with once every two weeks.

Other

Need to have a community for sharing and discussing.

Summary and Action Items

1. Build a community enabling the members to express their ideas/queries and the group can share their feedback. Couple of ways of forming the community will be

• Build a forum in [dspLog]
• Use some of the available social networking sites like FaceBook, Orkut etc

I think through and update you. In the meanwhile, you can continue asking your questions via comments and/or email. If the queries are relevant, I will post both the question and my response as an individual post and solicit feedback from the group.

2. Write more posts on receiver synchronization.

3. Discuss building blocks in various specifications like IEEE 802.11a, 802.16, Bluetooth etc.

4. I will stick to the current template for at least the next 3-4 months (last changed in May).

5. I am happy that the group is fine with the 1 post per week cycle. However, I will try and increase to two posts per week.

Once again, thanks a lot for your feedback. If you have any more comments, kindly do let me know.

Related posts:

1. dspLog turns two! Happy Birthday!
2. ICCBN 2008, July 17-20 2008, IISc, Bangalore
3. Join dspLog at Google FriendConnect

The definition of Toeplitz matrix from [1] is:

A matrix is said to be Toeplitz if the elements are determined completely by the difference .

Pictorially, if a line is drawn parallel to the main diagonal then all the elements on this line are equal.

A Toeplitz matrix is completely determined by the 1st row and 1st column of the matrix i.e. by elements.

For example, the matrix

is a Toeplitz matrix.

With this understanding, let us move on to some useful examples in Matlab where the Toeplitz matrix construction is used to implement some standard functions.

Implementing convolution

Convolution of two functions and is the sum of the product of one function with the time reversed copy of other function i.e.

.

Typically, one can visualize implementing this using a for-loop OR use the conv() function in MATLAB. However, if you don’t want to use the conv() operator and do not want the for-loop, you can still implement convolution by as a matrix multiplication by constructing one of input as a Toeplitz matrix.

% implementing convolution
x = [1:10]; % first sequence
h = [1 1 -1 2 3 1 -1]; % second sequence

% Forming the toeplitz matrix from x
% each column of the matrix stores the values of x as they slide in through the tapped delay line.

% the number of elements in the tapped delay line is equal to the number
% of elements in h

xM = toeplitz([x(1) zeros(1,length(h)-1) ], [x zeros(1,length(h)-1) ]);

y1 = conv(x,h); % convolution output

y2 = h*xM; % convolution using toeplitz matrix

diff = (y2 – y1)

Implementing moving average sum

Moving average sum of stores the mean of the latest N samples i.e.

.

Infact, the N point moving average sum can be implemented as convolution of the input sequence with a matrix having N ones. Since, the operation involves convolution, the same can be implemented using a Toeplitz matrix structure for the input.

% implementing Moving Average sum of the latest N samples
x = [1:10]; % input sequence
N = 5; % number of samples to be accumulated
% forming the toeplitz matrix from x
% each column of the matriz stores the latest N samples
xM = toeplitz([x(1) zeros(1,N-1) ], [x ]);
y1 = (1/N)*sum(xM);

y2 = conv(x,ones(1,N))/N; % implementing MA as convolutions

diff = y1 – y2(1:length(x))

Implementing auto correlation of the signal

The auto-correlation of the signal is defined as

.

Obviously as this involves convolution, the operation can be implemented as the product of two matrices, where one of them is a Toeplitz matrix.

x = cos(2*pi*1/32*[1:320]); % input sequence
N = length(x)

% padded x with N-1 zeros on both sides
xM = [ zeros(1,N-1) x zeros(1,N-1)];
% modified x to form a toeplitx matrix. The first column of x stores
% the elements of x followed by 2*N-2 zeros. The second column stores
% zero followed by x followed by 2*N-1 zeros and so on…
% the matrix stores x
xDelay_M =[ toeplitz([x zeros(1,2*N-2)], [ x(1) zeros(1,2*N-2) ] ) ];

y1 = xM*conj(xDelay_M);
y2 = xcorr(x);

% mean square error
diff = (y2-y1)*(y2-y1)’/length(y2-y1)

As can be observed, the output of the Toeplitx matrix implementation matches the xcorr() function provided in Matlab.

Implementing cross correlation of the signal

Once auto-correlation implementation is shown to be correct, it might be reasonably easy to extend it for the cross correlation case. The cross-correlation of two sequence and is

.

This can be implemented using the same steps performed for computing auto-correlation. However, to address the case where the size of the input sequence differ, the shorter sequence is padded with zeros.

x = 10*cos(2*pi*1/32*[1:320]) ;
y = randn(1,310) + j*randn(1,310)
nX = length(x);
nY = length(y);

% when the length of x, y are different, padding the smaller matrix with
% zeros
m = max(nX,nY);
x1 = zeros(1,m);
y1 = zeros(1,m);
x1(1:length(x)) = x;
y1(1:length(y)) = y;

% padded x with m-1 zeros on both sides
xM = [ zeros(1,m-1) x1 zeros(1,m-1)];
% modified y1 to form a toeplitx matrix.
yDelay_M =[ toeplitz([y1 zeros(1,2*m-2)],[ y1(1) zeros(1,2*m-2) ] ) ];
op1 = xM*conj(yDelay_M);

op2 = xcorr(x,y);
diff = (op1-op2)*(op1-op2)’/length(op1-op2);

It can be observed that the output of the xcorr() function in Matlab matches with the output from the Toeplitz implementation.

Implementing auto-correlation of the signal delayed by D samples and the accumulated by N samples

Typically, for identifying noisy sequence with a known periodic properties, it might be desirable to perform autocorrelation for a given delay D accumulated for given number of samples N. The operation is:

.

The sequence x is converted into two Toeplitz matrices – one storing the current samples and the other storing the delayed samples. The output is computed by taking the column sum of dot product multiplication of the matrix storing the current samples with the conjugate of the delayed sample matrix.

ip = cos(2*pi*1/10*[1:10]);

x = [ ip ip ip ip ip zeros(1,40) ] ; % input sequence
D = 10; % delay
N = 20; % accumulation

% forming the toeplitz matrix storing the delayed elements
% The first D columns of the matrix is all zeros. From D+1 column onwards,
% the elements on x starts sliding in. Each column is of length N
xDelay_M = toeplitz([zeros(1,N) ], [zeros(1,D) x zeros(1,N-1)]);

% forming the toeplitz matrix storing the current elements
% From 1st column onwards, the elements on x starts sliding in. Each column
% is of length N. The last D columns are all zeros.
xM = toeplitz([x(1) zeros(1,N-1) ], [x zeros(1,D + N-1)]);

% dot product of the current and the conjugate of the delayed sequence
y1 = sum(conj(xDelay_M).*xM);
plot(abs(y1));

As expected it can be seen that, due to the D delayed periodicity of the input, the auto-correlation output rises and hits the plateau value at the (D+N)th sample. The autocorrelation output starts falling from the plateau value at the end of the periodic seqeunce, which in this example is the 50th sample.

References:

[1] Multirate Systems And Filter Banks, P. P. Vaidyanathan

Technorati tags: Toeplitz matrix

Related posts:

1. Polyphase filters for interpolation
2. BER for BPSK in ISI channel with Zero Forcing equalization
3. Example of Cascaded Integrator Comb filter in Matlab