In the past, we had discussed BER for BPSK in flat fading Rayleigh channel and BER for BPSK in a frequency selective channel using Zero Forcing Equalization. In this post, lets discuss a frequency selective channel with the use of Minimum Mean Square Error (MMSE) equalization to compensate for the inter symbol interference (ISI). For simplifying the discussion, we will assume that there is no pulse shaping at the transmitter. The ISI channel is assumed to be a fixed 3 tap channel.

Transmit symbol

Let the transmit symbols be modeled as

, where

is the symbol period,

is the symbol to transmit,

is the transmit filter,

is the symbol index and

is the output waveform.

For simplicity, lets assume that the transmit pulse shaping filter is not present, i.e .

So the transmit symbols can be modeled by the discrete time equivalent

Figure: Transmit symbols

Channel Model

Lets us assume the channel to be a 3 tap multipath channel with spacing i.e.

Figure: Channel model (3 tap multipath)

In addition to the multipath channel, the received signal gets corrupted by noise , typically referred to as Additive White Gaussian Noise (AWGN). The values of the noise follows the Gaussian probability distribution function, with

mean and

variance .

The received signal is

, where

is the convolution operator.

MMSE Equalization

In Minimum Mean Square Error solution, for each sample time we would want to find a set of coefficients which minimizes the error between the desired signal and the equalized signal , i.e.

,

where,

is the error at sample time ,

is column vector of dimension storing the equalization coefficients,

is column vector of dimension storing the received samples,

is the number of taps in the equalizer,

is the cross correlation between received sequence and input sequence ,

is the cross correlation between received sequence and input sequence and

is the auto-correlation of the received sequence.

For solving the Minimum Mean Square Error (MMSE) criterion, we need to find a set of coefficients which minimizes .

Differentiation with respect to and equating to 0,

.

Simplifying,

,

Note :

a) is the variance of the input signal

b) (as there is no correlation between input signal and noise)

Simulation Model

Click here to download: Matlab/Octave script for computing BER for BPSK with 3 tap ISI channel with MMSE Equalization

The attached Matlab/Octave simulation script performs the following:

(a) Generation of random binary sequence

(b) BPSK modulation i.e bit 0 represented as -1 and bit 1 represented as +1

(c) Convolving the symbols with a 3-tap fixed fading channel.

(d) Adding White Gaussian Noise

(e) Computing the MMSE and ZF equalization filter at the receiver (with 7 taps in length)

(f) Demodulation and conversion to bits

(g) Counting the number of bit errors

(h) Repeating for multiple values of Eb/No

The simulation results are as shown in the plot below.

Figure: BER plot for BPSK in a 3 tap ISI channel with MMSE equalizer

Observations

1. Can see around 0.5dB gain with using MMSE equalizer

Reference

Complex to Real : Tutorial 26 – Filters, analog, digital and adaptive equalization

Related posts:

1. BER for BPSK in ISI channel with Zero Forcing equalization
2. MIMO with MMSE SIC and optimal ordering
3. MIMO with MMSE equalizer

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.

The article in DSPDesignline.com details the following:

• Based on the knowledge of bandwidth requirements for each type of modulation scheme, the capacity in bits/seconds/Hz is listed.

• Using the knowledge that the symbol to noise ratio is times the bit to noise ratio , the symbol error rate vs curves are plotted.
• Using symbol error rate versus plots, the required for achieving symbol error rate of is identified.
• Upon having the capacity and requirement, the requirements for BPSK, QPSK, 4PAM, 16QAM, 16PSK, 64QAM and 32PSK are mapped on to the Shannon’s capacity vs Eb/No curve.
• Assuming Gray coded modulation mapping, each symbol error causes one bit out of bits to be in error. So, the relation between symbol error and bit error is,.
• Using this assumption, the Bit Error Rate (BER) for BPSK, QPSK, 4PAM, 16QAM, 16PSK, 64QAM and 32PSK are listed and the BER vs Eb/No curve plotted.

Hope this article serves as a nice overview of the various digital modulation schemes. Click here to read the full article in DSPDesignline.com

Related posts:

1. Comparing 16PSK vs 16QAM for symbol error rate
2. Article in DSPDesignLine.com: M-QAM symbol error
3. 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

In the first post in this series, we have discussed Shannon’s equation for capacity of band limited additive white Gaussian noise channel with an average transmit power constraint. The capacity is,

bits/second

where

is the capacity in bits per second, is the bandwidth in Hertz, is the signal power and is the noise spectral density.

Capacity with increasing signal power

Increasing the signal power will mean that we can split the signal level into more number of levels even while ensuring low probability of error. Hence increasing signal power will lead to more capacity. However, as the increase in capacity is a logarithmic function of power, the returns are diminishing.


Matlab/Octave script for plotting capacity vs power
B=1;
N0=1;
P= [0:10^4];
C = B.*log2(1+P./(N0*B));
plot(P,C); xlabel('power, P'); ylabel('bandwidth,B'); ylabel('capacity, C bit/sec'); title('Capacity vs Power')

Figure: Capacity vs Power, keeping Noise and Bandwidth to unity

Can observe that increase in capacity is diminishing as we keep increase the value of power.

Capacity with increasing bandwidth

The second variable to play with is the bandwidth. Increasing the bandwidth has two effects:

1. More bandwidth means we can have more transmissions per second, hence higher the capacity.

2. However, more bandwidth also means that there is more noise power at the receiver.

The latter reduces the performance.

Let us try to evaluate the capacity equation when bandwidth tends to infinity i.e
.

From the Taylor series expansion, we know that

.

Applying this to the above equation,

.

This means that increasing bandwidth alone will not lead to increase of the capacity.

Matlab/Octave script for plotting capacity vs bandwidth
P = 1;
N0 = 1;
B = [1:10^3];
C = B.*log2(1+P./(N0*B));
plot(B,C)
xlabel('bandwidth, B Hz'); ylabel('capacity, C bit/sec'); title('Capacity vs Bandwidth')

Figure: Capacity vs Bandwidth, keeping signal power and noise power to unity

Can observe that the maximum achievable capacity by increasing bandwidth is 1.44 times the value.

Capacity (in bit/sec/Hz) vs Bit to noise ratio (Eb/No)

From our discussion till now, we have understood that a practical communication should have a rate which is lower than capacity , i.e.

bits/second.

Dividing both sides of the equation by bandwidth ,

bits/second/Hz.

Further, from our discussion on Bit error rate for 16PSK modulation using Gray mapping, we know that symbol to noise ratio is times the bit to noise ratio, i.e.
.

Substituting this into the capacity equation,
bits/second/Hz.

For notational convenience, let us define as the spectral efficiency in bits/second/Hertz.

The above equation can be equivalently represented as,

.

In the above equation, when tends to zero, the bit to noise ratio should be,

.

(Thanks to L’Hospital’s rule).

This means that for reliable communication, we need to have or equivalently expressing in decibels, .


Matlab/Octave script for plotting the capacity in Bits/sec/Hz vs Bit to noise ratio
r = [0:.001:10];
Eb_No_lin = (2.^r -1)./r;
Eb_No_dB = 10*log10(Eb_No_lin);
semilogy(Eb_No_dB,r)
axis([-2 20 0.1 10]); grid on
xlabel('Bit to noise ratio, Eb/No dB'); ylabel('Spectral efficiency, R/W bit/sec/Hz')
title('Spectral efficiency vs Bit to Noise ratio')


Figure: Spectral efficiency vs bit to noise ratio

The above plot captures the equation,

.

It divides the area into two regions:

(a) In the region below the curve, reliable communication is possible and

(b) in the region above the curve, reliable communication is not possible.

Closer the performance of a communication system is to the curve, more optimal is the system.

In the next post in this series, we will discuss the performance of various modulation schemes like BPSK, QPSK, QAM etc by mapping them into various points in the above plot.

Reference

[COMM-SYS-PROAKIS-SALEHI] Fundamentals of Communication Systems by John G. Proakis, Masoud Salehi

Related posts:

1. Understanding Shannon’s capacity equation
2. Comparing BPSK, QPSK, 4PAM, 16QAM, 16PSK, 64QAM and 32PSK
3. Sigma delta modulation

Let us try to understand the formula for Channel Capacity with an Average Power Limitation, described in Section 25 of the landmark paper A Mathematical Theory for Communication, by Mr. Claude Shannon.

Further, the following writeup is based on Section 12.5.1 from Fundamentals of Communication Systems by John G. Proakis, Masoud Salehi

Simple example with voltage levels

Let us consider that we have two voltage sources:
(a) Signal source which can generate voltages in the range to volts
(b) Noise source which can generate voltage levels to volts.

Figure: Discrete voltage levels with noise

Let us now try to send information at discrete voltage levels from the source (thick black lines as shown in the above figure). It is intuitive to guess that the receiver will be able to decode the received symbol correctly if the received signal lies within .

So, the number of different discrete voltages levels (information) which can be sent, while ensuring error free communication is the total voltage level divided by the noise voltage level i.e.
.

Extending to Gaussian channel

Let us transmit randomly chosen discrete voltage levels meeting the average power constraint,

, where is the signal power.

The noise signal follows the Gaussian probability distribution function

with mean and variance .

The noise power is,

.

The average total (signal plus noise) voltage over symbols is .

Similiarly, the average noise voltage over symbols is .

Combining the above two equations, the number of different messages which can be ‘reliably transmitted‘ is,

.

Note:

1. The product of the signal and noise accumulated over many symbols average to zero, i.e

.

2. Since the noise is Gaussian distributed, the noise can theoretically go from to . So the above result cannot ensure zero probability of error in receiver, but only arbitrarily small probability of error.

Converting to bits per transmission

With different messages, the number of bits which can be transmitted per transmission is,

bits/transmission.

Bringing bandwidth into the equation

Let us assume that the available bandwidth is .

Noise is of power spectral density spread over the bandwidth . So the noise power in terms of power spectral density and bandwidth is,

.

From our previous post on transmit pulse shaping filter that minimum required bandwidth for transmitting symbols with symbol period isHz. Conversely, if the available bandwidth is , the maximum symbol rate (transmissions per second) is .

Multiplying the equation for bits per transmission with transmission per second of and replacing the noise term , the capacity is

bits/second.

Voila! This is Shannon’s equation for capacity of band limited additive white Gaussian noise channel with an average transmit power constraint.

References

A Mathematical Theory for Communication, by Mr. Claude Shannon

[COMM-SYS-PROAKIS-SALEHI] Fundamentals of Communication Systems by John G. Proakis, Masoud Salehi

Related posts:

1. Bounds on Communication based on Shannon’s capacity
2. Digital implementation of RC low pass filter
3. Solved objective questions (GATE)