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BER for BPSK in ISI channel with MMSE equalization

Krishna Sankar — Sun, 24 Jan 2010 06:34:09 +0000

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

(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

BER for BPSK in ISI channel with Zero Forcing equalization

Krishna Sankar — Sat, 28 Nov 2009 20:15:21 +0000

In the past, we had discussed BER for BPSK in flat fading Rayleigh channel. In this post, lets discuss a frequency selective channel with the use of Zero Forcing (ZF) 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)

mean and

variance .

The received signal is

, where

is the convolution operator.

Zero Forcing Equalization

Objective of Zero Forcing Equalization is to find a set of filter coefficients which can make .

After equalization

Note:

The term causes noise amplification resulting poorer bit error rate performance.

Deriving the equalization coefficients

From the post on toeplitz matrix,we know that convolution operation can be represented as matrix multiplication.

% Matlab code for using Toeplitz matrix for convolution
clear all
x = [1:3];
h = [4:6];
xM = toeplitz([x zeros(1,length(h)-1) ], [x(1) zeros(1,length(h)-1) ]);
y1 = xM*h';
y2 = conv(x,h);
diff = y1'-y2
diff = [  0   0   0   0   0 ]

Using similar matrix algebra and assuming that the coefficients has 3 taps, the equation can be equivalently represented as,

Solving for , we have

If we assume that has 5 taps,

Solving for , we have

Example

% Assuming a 3 tap channel as follows
ht = [0.2 0.9 0.3];
L = length(ht);
kk = 1;
hM = toeplitz([ht([2:end]) zeros(1,2*kk+1-L+1)], [ ht([2:-1:1]) zeros(1,2*kk+1-L+1) ]);
d  = zeros(1,2*kk+1);
d(kk+1) = 1;
c  = [inv(hM)*d.'].';

The frequency response of the channel and the equalizer are shown below:

Figure: Frequency response of the channel and the equalizer

Simulation Model

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

(d) Adding White Gaussian Noise

(e) Computing the equalization filter at the receiver – the equalization filter is 3, 5, 7, 9 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 Zero Forcing equalizer

Observations

1. Increasing the equalizer tap length from 3 to 5 showed reasonable performance improvement.

2. Diminishing returns from improving the equalizer tap length above 5.

3. The results are poorer compared to the AWGN no multipath results. This is due to the noise amplification (see the frequency response above) by the zero forcing equalization filter.

Next step is to discuss the zero forcing equalizer in the presence of transmit pulse shaping and then move on to minimum mean square error equalizer.

BER with Matched Filtering

Krishna Sankar — Fri, 08 May 2009 00:40:18 +0000

In the post on transmit pulse shaping filter, we had discussed pulse shaping using rectangular and sinc. In this post we will discuss about optimal receiver structure when pulse shaping is used at the transmitter. The receiver structure is also called as matched filter. For the discussion, we will assume rectangular pulse shaping, the channel is AWGN only and the modulation is BPSK.

Transmitter

The simplified transmitter structure is as shown below:

Transmit pulse shaping filter

Figure: Transmit block diagram

The sequence of operation are as follows:

a). The random binary data comes as input to the the BPSK modulator circuit.

b). The BPSK modulator maps bits to symbols i.e bit0 to -1 and bit1 to +1

The sequence of transmit symbols maybe visualized as follows.

Figure: Transmit symbols for baseband PAM transmission

c). The upsampling block (the block with upward arrow) insert zeros between the samples based on the oversampling factor. For example, if the oversampling factor is 4, the block inserts 3 zeros between samples.

d). The last block in the chain is the filter, where the upsampled sequence is convolved with the filter. In this example, we have assumued rectangular filtering (also known as zero order hold), i.e. to repeat the current symbol till the next symbol arrives and so on. Mathematically, the filter can be represented as,

The filtered waveform can be as shown below.

Figure: Baseband PAM transmisison with rectangular filtering

The transmit spectrum assume that the symbol duration is shown below.

Figure: Transmit spectrum for rectangular matched filter

Matched filter Receiver

The filter having an impulse response is called a matched filter of .

Matched filter response for rectangular pulse

Figure: Matched filter response for a rectangular pulse shaping filter

In this example, as is a rectangular function, the mtched filter is also rectangular.

We pass the received signal through the matched filter . The output of the convolution peaks at time and we sample the at that time instant to perform the demodulation.

Further, if the filtered transmit signal is corrupted by noise, the filter with impulse response matched to maximizes the output Signal to Noise Ratio. The proof for this claim is detailed in Chapter 5.1.2 Matched Filter Demodulator in Digital Communications by John Proakis.

Simulation Model

The Matlab/Octave script performs the following

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

(b) Upsample by inserting zeros and the convolve with rectangular filter

(d) Convolve the received samples with matched filter, and extract output at time T

(e) Perform hard decision decoding and count the bit errors

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

Click here to download Matlab/Octave script for computing the BER for BPSK modulation in AWGN only with using rectangular pulse shaping at the transmitter and corresponding matched filter

Figure: BER plot for BPSK with rectangular pulse shaping and matched filtering

Observations

Can see that the simulated results are in good agreement with the theoretical BER results for BPSK in AWGN in AWGN.

References

Digital Communications by John Proakis.

Eye diagram with raised cosine filtering

Krishna Sankar — Thu, 01 May 2008 03:03:43 +0000

We have discussed about probable transmit pulse shaping filter and have observed that raised cosine filtering filtering allows a simpler implementation, albeit at the cost of increased bandwidth. Let us know understand the eye diagram, which is a useful graphical tool to quantify the degradation of the signal due to filtering.

Eye diagram

An eye diagram is generated in an oscilloscope operating in the persistence mode by observing the output of the filter with the symbol timing serving as the trigger. The observation window can be set as 2 times the symbol period. (Refer. Section 5.1.3 in [DIG-COMM-BARRY-LEE-MESSERSCHMITT]).

When the input data is random, the eye diagram which consists of many overlapped traces of the signal captures visually all the paths which the waveform takes.

Simulation script

Matlab/Octave script for simulating the eye diagram plot. The code performs the following

(a) Defines random BPSK modulated symbols (+1′s and -1′s)

(b) Defines two raised cosine filters with = 0.5, = 1

(d) Convolves the upsampled transmit sequence with the filter

(e) Overlays the time domain samples to plot the eye diagram

Click here to download: Matlab/Octave script for ploting the eye diagram

Update

25th May 2008

Corrected the issue. Modified the code to handle division by zero.

(a) for and

(b) for

(Thanks to the article in RFDesign.com, The care and feeding of digital, pulse-shaping filter, Ken Gentile)

19th May 2008

It has been brought to my attention that the code is unable to plot the eye diagram accurately in Matlab environment. The difference is because my version of Octave seems to handle the division by numbers close to zero cleanly, where as Matlab insists on returning Inf. I will fix the code and release an update. sorry for the inconvenience.

Figure: Eye diagram following raised cosine filtering with = 0.5

Figure: Eye diagram following raised cosine filtering with = 1

As can be observed from the above figures, the above waveform has a shape similar to the human eye and hence the name eye diagram.

Observations

1. For increasing the margin for error free transmission, the vertical opening of the eye should be more. In the presence of inter-symbol interference, the vertical opening of the eye reduces, thus increasing the probability of error.

2. The ideal sampling instant is the point where the vertical eye opening is maximum.

3. Smaller horizontal eye opening means implies more sensitivity to timing errors.

Note:

From the above figures, it can be observed that the horizontal eye opening with =0.5 is smaller than with =1.

Reason: The tails of the raised cosine filter with =1 dies away faster than the case where =0.5. Hence error in timing cause a bigger performance degradation for =0.5 than for=1 scenario. However, the flip side of using =1 is the increased bandwidth required for transmission.

Reference

[DIG-COMM-BARRY-LEE-MESSERSCHMITT] Digital Communication: Third Edition, by John R. Barry, Edward A. Lee, David G. Messerschmitt

Hope this helps.

Krishna

Raised cosine filter for transmit pulse shaping

Krishna Sankar — Tue, 22 Apr 2008 00:25:02 +0000

In the previous post on transmit filtering using Nyquist pulse, we had briefly learned that the information symbol with a symbol period can be transmitted without inter symbol interference (ISI) by using Nyquist pulse,

The resultant waveform is ideally bandlimited to frequencies from Hz to Hz.

However, in typical transmission schemes, we do not hear of pulse shaping using sinc() filters. Rather, pulse shaping using raised cosine filter is frequently used. In this post, objective is to understand the motivation behind using raised cosine filtering for pulse shaping.

Though the sinc filter achieves bandlimited transmission within Hz with out inter symbol interference, the sinc filter has the following issues:

1. The tail of the sinc filter decays slowly. Note that practical implementations cannot use a filter which extends from to . To ensure that only filter taps having small values are only ignored, need to use a filter of large length.

2. Small errors in timing synchronization at the receiver will result in significant intersymbol interference. Reason: The error in timing synchronization means that the sampling tme at the receiver is not aligned. This implies that filter tap values at time etc are non-zero. Hence reults in significant inter symbol interference.

Given so, there was a motivation to find filters which satisfies the Nyquist criterion, but has a faster decay of the filter tail. A commonly used pulse shaping filter satisfying the Nyquist criterion while having a faster decay is called the raised cosine filters having the following equation,

(Refer. Equation 5.8 in [DIG-COMM-BARRY-LEE-MESSERSCHMITT]).

where
is the excess bandwidth parameter and takes values from 0 to 1.

With =0, the raised cosine filter reduces to the classical Nyquist filter with zero excess bandwidth outside .

With =1 it is called 100% excess bandwidth and does not occupy frequencies outside .

Note:

(a) for and

(b) for

(Thanks to the article in RFDesign.com, The care and feeding of digital, pulse-shaping filter, Ken Gentile)

The frequency response of the raised cosine filter is,

Simulation model

Using the attached Matlab/Octave script, one can plot the time domain and frequency domain representations of the raised cosine filters for different values of .

Click here to download:

Matlab/Octave code for ploting the time and frequency response of raised cosine filter

Update

25th May 2008

Modified the code to handle the divison by zero error.

19th May 2008

It has been brought to my attention that the code is unable to plot accurately in Matlab environment. The difference is because my version of Octave seems to handle the division by numbers close to zero cleanly, where as Matlab insists on returning Inf. I will fix the code and release an update. sorry for the inconvenience.

Figure: Time domain response of raised cosine pulse shaping filters

Figure: Frequency domain response of raised cosine pulse shaping filters

Observations

[PROS] From the time domain samples, can observe that filter tail of the raised cosine filter with greater than 0 dies down faster. This implies that practical implementations can ignore taps which are close to zero with negligible loss in performance.

[PROS] As the filter taps values at and above are close to zero, timing mis-alignmnet at the receiver does not contribute to significant inter symbol interference.

[CONS] As can be seen from the frequency response, with greater than 0, the fitler response is bandlimited only till . We need a wider bandwidth to transmit the waveform when compared to classical Nyquist bandwidth.

Given that the raised cosine filtering simplifies the practical implementation (by making the receiver more robust to timing synchronization errors), the increase in transmission bandwidth may be a small price to pay.

Note

From the frequency domain response, one may observe that the shape of the roll-off looks like a cosine waveform having a DC value. Henec the term raised cosine filters.

Reference

[DIG-COMM-BARRY-LEE-MESSERSCHMITT] Digital Communication: Third Edition, by John R. Barry, Edward A. Lee, David G. Messerschmitt

Hope this helps.

Krishna

Transmit pulse shaping filter – rectangular and sinc (Nyquist)

Krishna Sankar — Mon, 14 Apr 2008 01:06:03 +0000

In the previous post on I-Q modulator and de-modulator, we had briefly mentioned that the a baseband PAM transmission can be modelled as

, where

is the symbol period,

is the symbol to transmit,

is the transmit filter,

is the symbol index and

is the output waveform.

In this post, the objective is to understand the properties of the transmit filter i.e. to find out a filter which occupies the minimum required bandwidth while ensuring inter-symbol-interference (ISI) free transmission of the information symbol .

The sequence of transmit symbols maybe visualized as follows.

Figure: Transmit symbols for baseband PAM transmission

To recover the symbols from , one may sample the waveform at multiples of symbol interval . The sampled waveform can be,

Breaking the above equation into two parts,

From the above equation, it is intuitive that for ensuring no inter symbol interference (ISI), the second term in the above equation should be zero i.e. the taps of the filter should be zero at and non-zero at time .

Rectangular filter

The most simplest filter is the zero-order hold filter, i.e. to repeat the current symbol till the next symbol arrives and so on. Mathematically, the filter can be represented as,

The filtered waveform can be as shown below.

Figure: Baseband PAM transmisison with rectangular filtering

Though there is no ISI with rectangular filtering, we will show later that this filtering is not be optimal from the bandwidth perspective.

Filtering with sinc() shaped pulses

From our textbooks, we may recall that sinc shaped pulses have have a band limited rectangular spectrum. For example, consider the sinc pulse of width .

Figure: Time domain response of sinc filter

As desired, the above filter has zeros at and non-zero value at time . The corresponding frequency response is a rectangular pulse bandlimited from Hz to Hz. The frequency response is as shown below:

Figure: Frequency response of sinc shaped filter

Infact, the above sinc shaped pulse satisfies the Nyquist criterion and is called Nyquist pulse (Refer Sec 5.1.1 [DIG-COMM-BARRY-LEE-MESSERSCHMITT]).

It follows that with the sinc shaped pulse (also called Nyquist pulse) used for transmit filtering:

(a) inter-symbol interference (ISI) is not introduced when sampled properly.

(b) minimum required bandwidth for transmitting symbols (with symbol period )isHz.

Simulation model

Brief Matlab/Octave script for observing the spectrum of a random BPSK modulated symbols with rectangular filtering and sinc shape filtering might be helpful to uderstand the concept further.

Click here to download.

Matlab/Octave code for simulating transmit pulse shaping with rectangular and sinc waveforms

The observed spectrum is as shown below.

Figure: Transmit spectrum with rectangular and sinc shaped pulse shaping filter

Summary

1. From the above spectrum plot, the sinc shaped filter does not result in perfectly bandlimited spectrum from from Hz to Hz. This is because, by theory, the sinc pulse exists from to . However, for simulations the sinc pulse is truncated for a finite duration. This results in negligible (less than -30dB) spectral content present outside Hz.

2. The above discussion does not present why the sinc shaped filtering is optimal for ensuring minimum bandwidth. For the theoretical understanding, one may refer Sec 5.1.1 of [DIG-COMM-BARRY-LEE-MESSERSCHMITT].

Reference

[DIG-COMM-BARRY-LEE-MESSERSCHMITT] Digital Communication: Third Edition, by John R. Barry, Edward A. Lee, David G. Messerschmitt

Hope this helps.

Krishna

Digital implementation of RC low pass filter

Krishna Sankar — Sat, 01 Dec 2007 23:39:40 +0000

Thanks to the nice article from Xilinx TechXclusives [XLNX-TECH], let us try to understand the probable digital implementation of resistor-capacitor based low pass filter. Consider a simple RC filter shown in the figure below. Assuming that there is no load across the capacitor, the capacitor charges and discharges through the resistor path.

Figure: RC low pass filter

is the voltage at the input of resistor and

is the voltage at the output.

From capacitor theory, the charge in the capacitor is , where

is the capacitance

is the voltage and

is the constant current flowing for short duration of time .

With the input voltage is greater than output voltage , resulting in current flowing through the resistor , where

When this current flows into the capacitor for a short time , the capacitor will charge and the voltage across the capacitor increases to

, where

is the new value of output voltage.

Digital Implementation

The above equation seems to be convenient for digital implementation as shown in the equation below:

where,

is ,

is and

is .

The transfer function of the above equation is

Simulation model

Script for plotting the frequency and step response of a digital RC low pass filter.

Click here to download.

Figure: Frequency response of the digital implementation of RC low pass filter

Figure: Step response of the digital implementation of RC low pass filter

Observations

1. As expected, lower the value of k, tighter is the frequency response and slower is the settling time. This is in synch with analog RC implementation, where a higher value of R and C suggests that the capacitor takes more time to charge/discharge (Note that k is inversely proportional to RC).

2. To facilitate simpler digital implementations not involving multipliers, the value of k can be chosen to be a power of 2.

References

[XLNX-TECH] Xilinx TechXclusives: Digitally Removing a DC Offset (or “DSP Without Math?”), By Ken Chapman Senior Staff Engineer, Applications Specialist, Xilinx UK.

Hope this helps.

Krishna