In this post, let us evaluate the impact of frequency offset resulting in Inter Carrier Interference (ICI) while receiving an OFDM modulated symbol. We will first discuss the OFDM transmission and reception, the effect of frequency offset and later we will define the loss of orthogonality and resulting signal to noise ratio (SNR) loss due to the presence of frequency offset. The analysis is accompanied by Matlab/Octave simulation scripts.

OFDM transmission

As discussed in the post on Understanding an OFDM transmission,  for sending an OFDM modulated symbol, we use multiple sinusoidals with frequency separation is used, where is the symbol period. The information to be send on each subcarrier is multiplied by the corresponding carrier and the sum of such modulated sinusoidals form the transmit signal. Mathematically, the transmit signal is,

The interpretation of the above equation is as follows:
(a) Each information signal multiplies the sinusoidal having frequency of .
(b) Sum of all such modulated sinusoidals are added and the resultant signal is sent out as .

OFDM reception

In an OFDM receiver, we will multiply the received signal with a bank of correlators and integrate over the period . The correlator to extract information send on subcarrier  is.

The integral,
,

where
takes values from till .

Frequency offset

In a typical wireless communication system, the signal to be transmitted is upconverted to a carrier frequency prior to transmission. The receiver is expected to tune to the same carrier frequency for down-converting the signal to baseband, prior to demodulation.

Figure: Up/down conversion

However, due to device impairments the carrier frequency of the receiver need not be same as the carrier frequency of the transmitter. When this happens, the received baseband signal, instead of being centered at DC (0MHz), will be centered at a frequency , where
.

The baseband representation is (ignoring noise),

, where

is the received signal

is the transmitted signal and

is the frequency offset.

Effect of frequency offset in OFDM receiver

Let us assume that the frequency offset is a fraction of subcarrier spacing i.e.
.

Also, for simplifying the equations, lets us assume that the transmitted symbols on all subcarriers, 

The received signal is,

.

The output of the correlator for sub-carrier is,

.

For  ,

The integral reduces to the OFDM receiver with no impairments case.

However for non zero values of , we can see that the amplitude of the correlation with subcarrier includes

• distortion due to frequency offset between actual frequency  and the desired frequency .
• distortion due to interference with other subcarriers with with desired frequency . This term is also known as Inter Carrier Interference (ICI).

Simulation Model

Click here to download the Matlab/Octave script for computing RMS error with frequency offset. The script performs the following:

1. Generates an OFDM symbol with all subcarriers modulated with .

2. Introduce frequency offset and add noise to result in =30dB.

3. Perform demodulation at the receiver.

4. Find the difference between the desired and actual constellation.

5. Compute the rms value of error across all subcarriers.

6. Repeat this for different values of frequency offset.

Figure: Error Magnitude vs frequency offset for OFDM

Observations

The theoretical results are computed by .

Quite likely the simulated results are slightly better than theoretical results because the simulated results are computed using average error for all subcarriers (and the subcarriers at the edge undergo lower distortion).

Related posts:

1. Frequency offset estimation using 802.11a short preamble
2. Understanding an OFDM transmission
3. Peak to Average Power Ratio for OFDM

In the post on Soft Input Viterbi decoder, we had discussed BPSK modulation with convolutional coding and soft input Viterbi decoding in AWGN channel. Let us know discuss the derivation of soft bits for 16QAM modulation scheme with Gray coded bit mapping. The channel is assumed to be AWGN alone.

Gray Mapped 16-QAM constellation

In the past, we had discussed BER for 16QAM in AWGN modulation. The 4 bits in each constellation point can be considered as two bits each on independent 4-PAM modulation on I-axis and Q-axis respectively.

Table: Gray coded constellation mapping for 16-QAM

Figure: 16QAM constellation plot with Gray coded mapping

Channel Model

The received coded sequence is

, where

is the modulated coded sequence taking values in the alphabet

.

is the Additive White Gaussian Noise following the probability distribution function,

with mean and variance .

Demodulation

For demodulation, we would want to maximize the probability that the bit was transmitted given we received i.e . This criterion is called Maximum a posteriori probability (MAP).

Using Bayes rule,

.

Note: The probability that all constellation points occur are equally likely, so maximizing is equivalent to maximizing .

Soft bit for b0

The bit mapping for the bit b0 with 16QAM Gray coded mapping is shown below. We can see that when b0 toggles from 0 to 1, only the real part of the constellation is affected.

Figure: Bit b0 for 16QAM Gray coded mapping

When the b0 is 0, the real part of the QAM constellation takes values -3 or -1. The conditional probability of the received signal given b0 is 0 is,

.

When the bit0 is 1, the real part of the QAM constellation takes values +1 or +3. The conditional probability given b0 is zero is,

.

We can define a likelihood ratio that if

.

The likelihood ratio for b0 is,

.

Region #1 ()

When , then we can assume that relative contribution by constellation +3 in the numerator and -1 in the denominator is less and can be ignored. So the likelihood ratio reduces to,

.

Taking logarithm on both sides,

.

Region #2 (), Region #3 ()

When or , then we can assume that relative contribution by constellation +3 in the numerator and -3 in the denominator is less and can be ignored. So the likelihood ratio reduces to,

.

Taking logarithm on both sides,

.

Region #4 ()

If , then we can assume that relative contribution by constellation +1 in the numerator and -3 in the denominator is less and can be ignored. So the likelihood ratio reduces to,

.

Taking logarithm on both sides,

.

Soft bit for b1

The bit mapping for the bit b1 with 16QAM Gray coded mapping is shown below. We can see that when b0 toggles from 0 to 1, only the real part of the constellation is affected.

Figure: Bit b1 for 16QAM Gray coded mapping

When the b1 is zero, the real part of the QAM constellation takes values -3 or +3. The conditional probability given b0 is zero is,

.

When the bit0 is 1, the real part of the QAM constellation takes values -1 or +1. The conditional probability given b0 is zero is,

.

We can define a likelihood ratio that if

.

The likelihood ratio for b1 is,

.

Region #1 (), Region#2 ()

When or , then we can assume that relative contribution by constellation +1 in the numerator and +3 in the denominator is less and can be ignored. So the likelihood ratio reduces to,

.

Taking logarithm on both sides,

.

Region #3 (), Region #4 ()

If or , then we can assume that relative contribution by constellation -1 in the numerator and -3 in the denominator is less and can be ignored. So the likelihood ratio reduces to,

.

Taking logarithm on both sides,

.

Summary

Note: As the factor  is common to all the terms, it can be removed.

The softbit for bit b0 is,

.

The softbit for bit b1 is,

.

The softbit for bit b1 can be simplified to,

.

It is easy to observe that the softbits for bits b2, b3 are identical to softbits for b0, b1 respectively except that the decisions are based on the imaginary component of the received vector .

The softbit for bit b2 is,

.

The softbit for bit b3 is,

.

As described in the paper, Simplified Soft-Output Demapper for Binary Interleaved COFDM with Application to HIPERLAN/2, Filippo Tosato1, Paola Bisaglia, HPL-2001-246 October 10th , 2001, the term

and

,

This simplification avoids the need for having a threshold check in the receiver for sofbits b0 and b2 respectively.

Reference

Simplified Soft-Output Demapper for Binary Interleaved COFDM with Application to HIPERLAN/2, Filippo Tosato1, Paola Bisaglia, HPL-2001-246 October 10th , 2001

Related posts:

1. Binary to Gray code for 16QAM
2. 16QAM Bit Error Rate (BER) with Gray mapping
3. Bit error rate for 16PSK modulation using Gray mapping

Given that we have discussed Binary to Gray code conversion, let us discuss the Gray to BInary conversion.

Conversion from Gray code to natural Binary

Let be the equivalent Gray code for an bit binary number with representing the index of the bit.

1. For ,

i.e, the most significant bit (MSB) of the Gray code is same as the MSB of original binary number.

2. For ,

i.e, bit of the Binary number is the exclusive-OR (XOR) of of the bit of the Gray code and of the bit of the binary number.

Simulation model

Look Up Table based Matlab/Octave mode for Gray to Binary conversion

clear
% binary to gray code conversion
ipBin = [0:15] ; % decimal equivalent for a 4-bit binary
opGray = bitxor(ipBin,floor(ipBin/2))
% Gray to Binary conversion
[tt ind] = sort(opGray); % sorting Gray code elements to form the lookup table
opBin = ind(opGray+1)-1; % picking elements from the array

The Trick from DSP-Guru to do Gray to Binary conversion, thanks to Jerry Avins.

clear
% binary to Gray code
ipBin = [0:15]
opGray = bitxor(ipBin,floor(ipBin/2))
% Gray code to binary
opBin = bitxor(opGray,floor(opGray/2^8)) ;
opBin = bitxor(opBin,floor(opBin/2^4)) ;
opBin = bitxor(opBin,floor(opBin/2^2)) ;
opBin = bitxor(opBin,floor(opBin/2^1)) ;

Concluding thoughts

1. As on now, I do not understand how the DSP Guru trick for Gray code to Binary works. Of course, if I figure out, I will update this post.

2. Having discussed Binary to Gray code conversion and Gray to Binary conversion, we are now armed to discuss the bit-error rate probabilities for various modulation schemes (Recall: We have been discussing symbol error rate in Additive White Gaussian noise till date).

Related posts:

1. Binary to Gray code for 16QAM
2. Binary to Gray code conversion for PSK and PAM
3. Bit error rate for 16PSK modulation using Gray mapping

Some time back, a friend mentioned that he is yet to understand the need for having both in-phase (I) and quadrature-phase (Q) signals in typical wireless systems.

In this post, the attempt is to bring out the motivation for having I-Q modulation and present the block diagram of a simple I-Q modulator (and demodulator).

I am using the text provided in Sec 5.2.1 of [DIG-COMM-BARRY-LEE-MESSERSCHMITT] as reference.

Baseband PAM transmission

Consider a simple baseband transmission where the information is sent by modulating a pulse. This can be represented as

, where

is the symbol period,

is the symbol to transmit,

is the transmit filter,

is thesymbol index and

is the output waveform.

Pictorially the same can be represented as,

Figure: Baseband PAM transmission (Refer: Figure 5.1 [DIG-COMM-BARRY-LEE-MESSERSCHMITT]]

To transmit undistorted through the channel, the minimum bandwidth required is half the symbol rate (per the Nyquist criterion).

Figure: Minimum bandwidth required for transmitting baseband PAM with symbol rate

Assuming that there are alphabets in , the spectral efficiency for basedband PAM is,

bits/second/Hertz.

BB-PAM : Baseband PAM

Passband PAM transmission

Now, consider that the baseband PAM signal is upconverted to a carrier frequency by multiplication by a carrier, i.e.

.

The spectrum of the passband PAM is as shown below,

Figure: Spectrum of passband PAM

To avoid transmit undistorted through the channel, we require a passband filter having bandwidth .

Assuming that there are alphabets in , the spectral efficiency for passband PAM is,

bits/second/Hertz.

PB-PAM : PassBand PAM

Can see that spectral efficiency of passband PAM is half of baseband PAM.

Note:

Passband PAM is composed of real signals and real signals have symmetric spectra. So, half the bandwidth is not carrying any ‘extra’ information.

Moving to passband QAM

It is known that if a sine wave and a cosine wave is periodic over time , then they are orthogonal i.e.
.

Given so, a popular way which people came across to improve the efficiency of the passband PAM is to send information on the sine wave also in parallel, i.e.

.

This forms the passband QAM modulation. This signal requires the same bandwidth as passband PAM, however carries twice the informtion. For notational convineance, the above signal can be represented mathematically as,

, where

and

.

This forms the I-Q modulator circuit.

Figure: IQ modulator

The IQ demodulator can be visualized as shown in the figure below

Figure: IQ demodulator

On the combined signal, downconvert the cosine (inphase, I) and the sine (quadrature Q) arms, then proceed to do independent demodulation on each arm.

Summarizing,

(a) It is the need for improved spectral efficiency that resulted in I-Q modulation (and de-modulation).

(b) Another approach for improving the spectral efficiency of passband PAM is to filter away half the bandwidth (which is not carrying any ‘extra’ information). This is called Single Sideband Modulation (SSB).

Anyhow, it seems that using I-Q modulation is simpler to implement than circuit for filtering away half the spectrum. So, I-Q modulation stays.

Reference

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

Thanks,
Krishna

Related posts:

1. Bounds on Communication based on Shannon’s capacity
2. Transmit pulse shaping filter – rectangular and sinc (Nyquist)
3. Understanding Shannon’s capacity equation

From the previous post on OFDM (here), we have understood that an OFDM waveform is made of sum of multiple sinusoidals (also called subcarriers) each modulated independently. In this post, let us try to understand the estimation of frequency offset in a typical OFDM receiver (using the short preamble specified per IEEE 802.11a specification as a reference).

Understanding frequency offset

In a typical wireless communication system, the signal to be transmitted is upconverted to a carrier frequency prior to transmission. The receiver is expected to tune to the same carrier frequency for downconverting the signal to baseband, prior to demodulation.

Figure: Up/down conversion

However, due to device impairments the carrier frequency of the receiver need not be same as the carrier frequency of the transmitter. When this happens, the received baseband signal, instead of being centered at DC (0MHz), will be centered at a frequency , where
.

The baseband representation is (ignoring noise),

, where

is the received signal

is the transmitted signal and

is the frequency offset.

Frequency offset estimation in 802.11a OFDM preamble

From the IEEE 802.11a specifications (Sec 17.3.3), it can be observed that each OFDM packet has a preamble structure formed using 10 short preambles of duration each. This short preamble is constructed by defining 12 subcarriers only (out of the available 52 subcarriers) where the modulation of individual subcarriers ensure a low peak to average power ratio.

Figure: OFDM short preamble 802.11a specification

From the equation defined in the previous section,

.

Given that short preamble is perodic with ,

.

At the receiver as both and are known,

.

Taking angle() of both sides of the equation

.

So, the frequency offset is,

.

Simulation model

Simple Matlab/Octave script simulating generation of 802.11a short preamble, introducing frequency offset of 200kHz and estimating frequency offset is provided. Click here to download.

Figure: Plot of frequency offset estimate using 802.11a short preamble

As can be observed starting from sample number 17 (i.e. after ) and till samples number 160 () the frequency offset estimate is available. For improving accuracy in the presence of noise, typically the output is accumulated prior to computation of angle.

Maximum possible frequency offset which can be estimated

It is known that limits of the phase estimated by the angle() function is from . Plugging this to the above equation, the minimum value of frequency offset value which can be estimated is,

and the maximum value is,

.

From Sec17.3.9.4 (of IEEE 802.11a specification), the center frequency tolerance is . With a carrier frequency of 5.8GHz, this specification corresponds to a frequency offset within the range . Thankfully, this is within the ‘estimatable’ range of frequency offsets which which can be estimated by the short preamble .

Note:
We have not discussed the effect of inter carrier interference (ICI) between the sub-carriers which happens in the presence of frequency offset. Maybe we can discuss in a later post.

References

[802.11A] Wireless LAN Medium Access Control (MAC) and Physical (PHY) Layer specifications – High speed physical layer in 5GHz band

Hope this helps.
Krishna

Related posts:

1. Inter Carrier Interference (ICI) in OFDM due to frequency offset
2. Back!
3. Cylcic prefix in Orthogonal Frequency Division Multiplexing

Considering a typical scenario where there might exist a small phase offset between local oscillator between the transmitter and receiver.

Figure 1 :

Transmitter receiver with constant phase offset

In such cases, it might be desirable to estimate and track the phase offset such that the performance of the receiver does not degrade.

A simple first order digital phase locked loop for tracking the constat phase offset can be as

Assuming that the transmitter signal gets rotated by a constant phase , the received signal . In a simple no-noise case, assuming that the phase offset is small (and the signal gets decoded correctly), the estimate of phase offset is,

.

Typically, a first order phase locked loop which converges to is used for facilitating synchronous demodulation.

Figure 2 :

First order digital phase locked loop (PLL)

(adapted from Fig 5.7 of [Mengali])

The estimate from each sampling instant is accumulated to form the estimate . This estimated phase is removed from the received samples to generate . The parameteris a non-zero positive constant in the range controls the rate of convergence of the loop. Higher value of indicates faster convergence, but is more prone to noise effects. Lower value of is less noisy, but results in slower convergence.

Assuming , the phase estimate at the output of the filter is

.

Substituting, , then

.

A simple Matlab code to simulate this can be as follows:

% random +/-1 BPSK source
xn = 2*(rand(1,1000) >0.5) – 1;
% introducing a phase offset of 20 degrees
phiDeg = 20;
phiRad = phiDeg*pi/180;
yn = xn*exp(j*phiRad);
% first order pll
alpha = 0.01;
phiHat = 0;
for ii = 1:length(yn)
yn(ii) = yn(ii)*exp(-j*phiHat);
% demodulating circuit
xHat = 2*real(yn(ii)>0) -1 ;
phiHatT = angle(conj(xHat)*yn(ii));
% accumulation
phiHat = alpha*phiHatT + phiHat;
% dumping variables for plot
phiHatDump(ii) = phiHat;
end

plot(phiHatDump*180/pi,’r.-’)

Figure 3: Convergence of for two values.

Reference:

[1] Synchronization Techniques for Digital Receivers (Applications of Communications Theory) by Umberto Mengali

Related posts:

1. Frequency offset estimation using 802.11a short preamble
2. Zero-order hold and first-order hold based interpolation
3. 2nd order sigma delta modulator