This is the first post in the series discussing receiver diversity in a wireless link. Receiver diversity is a form of space diversity, where there are multiple antennas at the receiver. The presence of receiver diversity poses an interesting problem – how do we use ‘effectively‘ the information from all the antennas to demodulate the data. There are multiple ways to approach the problem. The three typical approaches to be discussed are – selection diversity, equal gain combining and maximal ratio combining. In this post we will discuss selection diversity. For the discussion, we will assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK.
Year: 2008
BER for BPSK in OFDM with Rayleigh multipath channel
Mr. Lealem Tamirat, in a comment on BER for BPSK in Rayleigh channel, wondered about the performance of an OFDM modulated system in a frequency selective Rayeligh fading channel. My response was that,
Though the total channel is a frequency selective channel, the channel experienced by each subcarrier in an OFDM system is a flat fading channel with each subcarrier experiencing independent Rayleigh fading.
So, assuming that the number of taps in the channel is lower than the cyclic prefix duration (which ensures that there is no inter symbol interference), the BER for BPSK with OFDM in a Rayleigh fading channel should be same as the result obtained for BER for BPSK in Rayleigh fading channel.
Let us try to define a quick simulation to confirm the claim.
Continue reading “BER for BPSK in OFDM with Rayleigh multipath channel”
Receive diversity in AWGN
Some among you will be aware that in a wireless link having multiple antenna’s at the receiver (aka receive diversity) improves the bit error rate (BER) performance. In this post, let us try to understand the BER improvement with receive diversity. And, since we are just getting started, let us limit ourselves to additive white Gaussian noise (AWGN) channel (i.e assume that the channel gains are unity).
BER for BPSK in Rayleigh channel
Long back in time we discussed the BER (bit error rate) for BPSK modulation in a simple AWGN channel (time stamps states August 2007). Almost an year back! It high time we discuss the BER for BPSK in a Rayleigh multipath channel.
In a brief discussion on Rayleigh channel, wherein we stated that a circularly symmetric complex Gaussian random variable is of the form,
,
where real and imaginary parts are zero mean independent and identically distributed (iid) Gaussian random variables with mean 0 and variance .
Negative Frequency
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).
Summary – feedback on [dspLog], July 2008
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.
Continue reading “Summary – feedback on [dspLog], July 2008”
OT: Happy Schools Blog
Mr. Raghuram contacted me and informed about Happy Schools Blog.
He writes about Graduate School Admission in U.S., Job opportunities for students, University Selection based on his personal experience.
He recently published few articles which might of interest to some of our readers. Here are the URL for few articles:
Chi Square Random Variable
While trying to derive the theoretical bit error rate (BER) for BPSK modulation in a Rayleigh fading channel, I realized that I need to discuss chi square random variable prior.
What is chi-square random variable?
Let there be independent and identically distributed Gaussian random variables
with mean
and variance
and we form a new random variable,
.
Then is a chi square random variable with
degrees of freedom.
ICCBN 2008, July 17-20 2008, IISc, Bangalore
Advanced Computing and Communication Society (ACS) of India is organizing ICCBN 2008 conference (International Conference on Communication, Convergence, and Broadband Networking) from July 17th to 20th 2008 at National Science Seminar Complex at Indian Institute of Science (IISc), Bangalore.
ICCBN Conference aims to provide a premier forum for researchers, industry practitioners and educators to present and discuss the most recent ideas, innovations, trends, experiences, and concerns in the emerging areas of Communication, Convergence, and Broadband Networking.
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Deriving PDF of Rayleigh random variable
In the post on Rayleigh channel model, we stated that a circularly symmetric random variable is of the form , where real and imaginary parts are zero mean independent and identically distributed (iid) Gaussian random variables. The magnitude
which has the probability density,
is called a Rayleigh random variable. Further, the phase is uniformly distributed from
. In this post we will try to derive the expression for probability density function (PDF) for
and
.
Rayleigh multipath channel model
The article gives a quick overview of a simple statistical multipath channel model called Rayleigh fading channel model.
Multipath environment
In a multipath environment, it is reasonably intuitive to visualize that an impulse transmitted from transmitter will reach the receiver as a train of impulses.
Comparing BPSK, QPSK, 4PAM, 16QAM, 16PSK, 64QAM and 32PSK
I have written another article in DSPDesginLine.com. This article can be treated as the third post in the series aimed at understanding Shannon’s capacity equation.
For the first two posts in the series are:
1. Understanding Shannon’s capacity equation
2. Bounds on Communication based on Shannon’s capacity
The article summarizes the symbol error rate derivations in AWGN for modulation schemes like BPSK, QPSK, 4PAM, 16QAM, 16PSK, 64QAM and 32PSK.
Continue reading “Comparing BPSK, QPSK, 4PAM, 16QAM, 16PSK, 64QAM and 32PSK”
Trying out PAPR reduction for OFDM by multiplication with j
In this post, we will explore a probable way of reducing PAPR (peak to average power ratio) in OFDM by changing the phase of some of the subcarriers. This is in response to the comment to post on Peak to Average power ratio for OFDM, where Mr. Elibom suggested to reduce the PAPR by cyclically rotate some of the subcarriers and using.
Further, the presentation in the IEEE TGN, PAPR in HT-LTF (11-06/1595r1), mentions that in 40MHz mode where a 128pt FFT is used, PAPR of HT-LTF (High Throughput Long Training Field) can be reduced by multiplying the upper 20MHz subcarriers by j. Using quick Matlab simulations, we will try to validate that claim for HT-LTF and further check the PAPR for a general random BPSK and QPSK modulation.
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Bounds on Communication based on Shannon’s capacity
This is the second post in the series aimed at developing a better understanding of Shannon’s capacity equation. In this post let us discuss the bounds on communication given the signal power and bandwidth constraint. Further, the following writeup is based on Section 12.6 from Fundamentals of Communication Systems by John G. Proakis, Masoud Salehi
Continue reading “Bounds on Communication based on Shannon’s capacity”