wavlet based ofdm

8/7/2019 Wavlet based OFDM

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EE678 WAVELETS APPLICATION ASSIGNMENT

WAVELET OFDM

GROUP MEMBERS

RISHABH KASLIWAL [email protected] 02D07001

NACHIKET KALE [email protected] 02D07002

PIYUSH NAHAR [email protected] 02D07007

WAVELET OFDM

In this paper we discuss the use of wavelets in OFDM.. Normally OFDM is

implemented using FFT and IFFTs. The FFT uses a rectangular window. A

rectangular window produces high side lobes. This causes interference when the

impairments are not compensated. In this paper we look at the replacement of

Fourier transform by wavelet transform and the restrictions imposed by this and

the condition on the wavelet construction. We look at the use of Wavelet Packet and

normal wavelet Transform and the use of bi-orthogonal wavelets.

Index TermsWavelet, OFDM, Wavelet Packet, FFT, Fourier Transform, OWPDM, DWT,

DWT-OFDM, DFT-OFDM

I INTRODUCTION

As the need for high speed communication grows, we turn to broadband communication.

Normally for a channel with small width, the frequency response is fairly flat throughout

the channel. Also, the noise in the channel is AWGN (Additive White Gaussian Noise).As the channel width grows, it is difficult to model the channel. Therefore we split the

channel into smaller sub-channels.Data transmission over a difficult channel is transformed through the use of advancedsignal processing techniques into the parallel transmission of the given data stream over a

large number of sub-channels such that each sub-channel may be viewed effectively as an

AWGN channel. Orthogonal Frequency division multiplexing offers an effective way tohandle high data rate. The OFDM requires a cyclic prefix to remove ISI. This causes

overhead and this overhead may be sometimes much large for the system to be effective.

In OWPDM the modulation and demodulation are implemented by wavelets rather than
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by Fourier transform. The use of wavelet promises to reduce the ISI and ICI. The wavelet

transform offers a higher suppression of side lobes. This paper is organized as follows:IN section II we discuss the basics of OFDM; In section III we discuss WAVELET AND

WAVELET PACKETS. In section IV we discuss the Wavelet OFDM and the system

model, In section V we discuss the channel model and the results; In section VI we

discuss the benefits of the Wavelet-OFDM.

II OFDM

Most of the modulation technique uses the advantage of flat channel. But with larger

channel width, this assumption is not valid. OFDM uses the policy of divide and rule. It

breaks down the problem of transmission over a large difficult channel into transmissionover smaller flat sub channels. The total incoming bit stream is divided over a large

number of sub channels. These bits are then modulated on each sub channel using

modulation technique like PSK, QAM etc. The sum of these composite sub carriers isthen sent over the channel. If the sub carriers are orthogonal then the different spectra

may overlap giving a larger spectral efficiency. In OFDM the transformation is

performed in discrete time as well as discrete frequency. The generation of the sub

carriers is done by using the Fourier transform. Let the number of sub carrier be Nc and mbe the number of bits that form a symbol for a sub carrier then OFDM input can be

considered as a block of Nc x m bits. The channel has a finite impulse response h(t)

confined to finite interval [0,Tb]. Let the sequence h1 h2 h3 hv denote the base bandequivalent impulse response of the channel sampled at rate 1/ TS, where Tb = (1+ v)TS.

The sampling rate is chosen to be greater than twice the higher frequency component of

interest.

Now consider the channel output x[n], input stream s[0] to s[N-1] as one symbol, andnoise w[n].They are related as:

x[n] = m s[m] h[n m]

Now the convolution of signal s of length N with channel impulse response h oflength v produces output sequence x of length N+v. This extension of the output

sequence by v is due to ISI. To overcome it, we use the concept of cyclically extended

guard interval. Instead of transmitting s[0],s[1],,s[N-1], we

transmits[N-v],s[N-v+1],s[N-1],s[0],s[1],s[N-1].This yields in the equation:

x[N-1] h[0] h[1] h[2] . h[v] 0 . 0 s[N-1] w[N-1]

x[N-2] 0 h[0] h[1] . h[v-1] h[v] . 0 s[N-2] w[N-2]

. = . . . .. . . + .

. . . . .. . . .

x[0] h[1] h[2] h[3] .. 0 0 .. h[0] s[0] w[0]


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The system response matrix H, called the circulant matrix, has the property of being able

to be expressed as H = Q` A Q where A is a diagonal matrix.Consider S` to be the frequency domain representation of s , the transmitter input where

s=Q` S`. Correspondingly the frequency domain representation of output x is given as

X` = Qx. Using the above matrix equation, we state

X` = Q ( Q` A Q Q` S + w )

Thus we simply get X` = AS` + W` where W` = Q w

Thus we can consider the data as frequency components and do an IFFT to get the timecomponents which are then transmitted over the channel; at the receiver we do a FFT to

get back the frequency component. This structure is represented as below.

Fig.1 Representation of the OFDM implementation


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III Wavelet Transform and Wavelet PACKETTransform

The wavelet transform is usually represented as MRA. The wavelet transformdecomposes the signal using a set of basis function into different resolution subspaces

..V-2< V-1 < V0 < V1


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Fig.2 The tree structure of the Wavelet decomposition

Fig 3. A Structure of wavelet decomposition.

The synthesis side can be considered as upsampling by 2 and then filtering the low andhigh pass coefficient at the k-th level and then adding the two and this gives the low pass

coefficient at the (k-1) level and similar structure at the subsequent levels give back the

signals.The filter of the synthesis side can be determined from the analysis side filter bythe perfect reconstruction condition. This gives a variety of filters and this leads to the

various families of wavelets. A 1st

level decomposition and reconstruction is shownbelow for QMF filters where and)()( nhh =n ).()( ngng =

Fig 4. A 1-level analysis and synthesis structure of the discrete wavelet transform


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In wavelet transform we do the decomposition of just the low pass coefficients. Ageneralization of this is the wavelet packet transform in which the decomposition is done

along both the high and low pass coefficients. Also we make use of the identities that

when a signal is downsampled and then passed through a filter (H(z)) and it is equivalent

to passing the signal through a filter(H(z

2

)) and then downsampling. Thus the waveletpacket transform can be represented as below. The frequency time plot of Wavelet

transform and wavelet packet transform is shown below. We see that at low frequencies

the time-span is larger while at high frequencies the time-span is smaller. For the waveletpacket we can decide how to decompose the high and low frequencies parts as after each

decomposition we can decide whether to decompose the signal in the low/high frequency

domain or not.

Fig. 5 Wavelet Packet Decomposition

IV Wavelet-OFDM or OPDWM

The OFDM implemented by using IFFTs and FFTs have some problems. The OFDM

suffers from

ISI (intersymbol interference) This is usually taken care of by using a adding acyclic prefix greater than the channel length but this may not always be possible.This occurs due to loss of orthogonality due to channel effects.


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Time and Frequency Synchronization- The OFDM requires time and frequencysynchronization to get a low bit error rate.

Carrier Frequency Offset- The offset between the carrier frequency and thefrequency of the local oscillator also causes a large bit error rate.

Due to these problems we need to look at other type of modulation to generate the carrier.One of these is the wavelet transform. The wavelet transform is proposed by many

authors, it has a higher degree of side lobe suppression and the loss of orthogonality leadsto lesser ISI and ICI. In Wavelet OFDM the FFT and IFFT is replace by DWT and IDWT

respectively. For the Wavelet transform we see that from the time-frequency plot that the

basic Wavelet transform offers lesser flexibility than the wavelet packet transform. Forthe wavelet packet transform we can construct an algorithm to do the decomposition such

that the effect due to the noise (assuming that we know the frequency that is affected

most by the noise and the time when it affected most).

Fig.6 The wavelet decomposition of a carrier.[2]

The above fig shows the case, the dark lines represent the noise. The top-left corner gives

the normal wavelet decomposition. The bottom-left shows a carrier with nodecomposition .The top-right shows a symmetrical tree structure, while the bottom-right

shows the optimum subcarrier decomposition. A subcarrier corresponds to a rectangle in

the time-frequency tiling. Thus if the maximum resolution is defined we can have the

same number of subcarrier regardless of the tree structure.

The wavelet based OFDM can be modeled as follows [4]:

The transmitter and receiver are shown in the figure below. At the transmitter the data is

first M-ary modulated and then serial to parallel converted first then upsampled and thenpassed through an IDWPT filter bank. At the receiver the data is passed through an

Analysis Filter bank and then parallel to serial converted and then M-ary demodulated


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Fig.7 The transmitter structure of the Wavelet Packet OFDM

Fig.8. The receiver structure of the Wavelet Packet OFDM

This is the case for wavelet packet transform, some researchers have also used wavelet

transform. In such a case the IFFT and FFTs in the OFDM are just replaced by the

DWT/IDWT to give the DWT-OFDMThe wavelet can be implemented using QMF filter banks but this cannot fulfill linear

phase filtering, therefore some authors[4] have suggested the use of bi-orthogonal filter

banks, bi-orthogonal filters provide linear phase filtering and the design is flexible.The design of the transmitter and receiver based on biorthogonal filter banks is similar to

the above design with the filters being the bi-orthogonal filters.

V Channel Model

We consider two channel models, an AGWN channel model [4] and a Rayleigh model

[4], [5].

1. AWGN channel:


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For an AWGN channel the parameters used are 4 subcarriers, BPSK

modulation and bi-orthogonal wavelets. In [4] it is shown that some of thewavelets perform worse than the single carrier case while ideally it should

perform the same. This can be explained as follows: the basis function for

a biorthogonal wavelet are not orthogonal to each other. This causes the

AWGN to become correlated within a subcarrier and thus an AWGNchannel doesnt remain an AWGN channel. Thus the non-orthogonality

has now become a problem, but some wavelets are still useful.

For the Haar wavelet case [5] the performance of a DWT-OFDM is muchbetter than the DFT-OFDM.case when no timing, frequency

synchronization is done.

Fig.9. The performance on an AWGN channel by bi-orthogonal wavelet

packets


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Fig.10 Performance of DFT-OFDM and DWT-OFDM over a AWGN

channel

2. Flat Rayleigh fading channelFor a Flat Rayleigh fading channel the channel can be modeled as follows:

Let the transmitted codeword be X and received be Y

Then Y=aX + n;Where a is the fading coefficient and is i.i.d. random variable with pdf

and n is the AWGN.

For the Flat Rayleigh channel it is found that the DWT-OFDM is

advantageous for some SNR ranges (SNR


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Fig 11. Performance of DFT-OFDM and DWT-OFDM over Rayleigh channel

The further advantages of the Wavelet OFDM are1. That it requires lesser overhead as it doesnt require CP2. It doesnt require a pilot tone which takes about 8% of the subbands.3. DWT-OFDM is inherently robust to ISI and ICI

VI Conclusion

We see that DWT-OFDM performs much better than the DFT-OFDM over AWGN and

Rayleigh channel with low SNR. Also we find that use of Wavelets reduces the

overhead thus giving a larger bandwidth. The wavelet packet is much better than theimplementing just the wavelet transform as it is more flexible. The bi-orthogonal

wavelets though may provide some advantage and better flexibility doesnt perform well

for some wavelets considered to the theoretical case.

References

1. Communication Systems, 4th edition, Simon Haykin, John Wiley and Sons,Inc..

2. C. V. Bouwel, et. al, Wavelet Packet Based Multicarrier Modulation, IEEECommunications and Vehicular Technology, SCVT 200, pp. 131-138, 2000

3. B. G. Nagesh, H. Nikookar, Wavelet Based OFDM for Wireless Channels, IEEEVehicular Technology Conference, Vol. 1, pp. 688-691, 2001.

4. LI Wiehua, et.al, Bi-orthogonal Wavelet Packet based Multicarrier modulation5. Haixia Zhang, et. al, Research of DFT-OFDM and DWT-OFDM on Different

Transmission Scenarios. Proceeding of the second international conference on

Information Technology for Application (ICITA 2004)


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wavlet based ofdm

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