EE678 WAVELETS APPLICATION ASSIGNMENT WAVELET OFDM GROUP MEMBERS RISHABH KASLIWAL rishkas@ee.iitb.ac.in 02D07001 NACHIKET KALE nachiket@ee.iitb.ac.in 02D07002 PIYUSH NAHAR nahar@ee.iitb.ac.in 02D07007 WAVELET OFDM In this paper we discuss the use of wavelets in OFDM.. Normally OFDM is implemented using FFT and IFFT s. 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 Terms Wavelet, 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 advanced signal 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 to handle 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
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 transmission over 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 is then 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 N c and m be the number of bits that form a symbol for a sub carrier then OFDM input can be considered as a block of N c x m bits. The channel has a finite impulse response h(t) confined to finite interval [0,T b ]. Let the sequence h 1 h 2 h 3 h v denote the base band equivalent impulse response of the channel sampled at rate 1/ T S, where T b = (1+ v)t S. 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, and noise 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 of length 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 transmit s[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]
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 time components 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
III Wavelet Transform and Wavelet PACKET Transform The wavelet transform is usually represented as MRA. The wavelet transform decomposes the signal using a set of basis function into different resolution subspaces..v -2 < V -1 < V 0 < V 1 < The decomposition is done using a basis function and a wavelet function and there translation and dilation. The dilated and translated scaling function forms the basis of the various subspaces. i.e. { ø (t)} forms a basis for V 0. The wavelet functions forms a subspace orthogonal to the basis formed by the scaling function. The scaling and the wavelet function both satisfy some dilation equation. ø(t) = ø(2t-n)h(n). If ø (t) should be orthonormal to its translated then h[n] should satisfy the orthonormality condition h[n]h[n-2m] = δ[m] and (-1) n h[n] =0 Given a sequence we can find another sequence g[n] such that the function satisfying the Dilation equation Ψ (t) = Ψ (2t-n)g[n]. This function is orthonormal to the scaling function is called the wavelet function. Using the wavelet and the scaling function we decompose the signal into two subspaces orthogonal to each other. Thus if the original signal is in space V 0 then using the scaling and wavelet function we decompose it into subspaces V 1 and W 1. In the classical wavelet transform the subspace V 1 is further decomposed into orthogonal subspaces V 2 and W 2. We see that ø(t) occupies only half the frequency space of ø(2t) and similarly for the wavelet function. Thus this decomposition can be considered as decomposition into high and low frequency domain. In discrete wavelet transform we can represent the process of decomposition as low and high pass filtering and then downsampling by 2. The filter coefficients are given by g[-n] and h[-n]. The filter with coefficients h[-n] forms a low pass filter while the filter with g[-n] forms a high pass filter. Thus the wavelet transform can be constructed by using QMF filter banks. The low passed and the high passed signals are down sampled by 2. The low pass signal can again be decomposed into high and low pass signals. This can be represented using a tree structure as shown below.
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 and high 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 by the perfect reconstruction condition. This gives a variety of filter s and this leads to the various families of wavelets. A 1 st level decomposition and reconstruction is shown below for QMF filters where h ( n) = h( n) and g( n) = g( n). Fig 4. A 1-level analysis and synthesis structure of the discrete wavelet transform
In wavelet transform we do the decomposition of just the low pass coefficients. A generalization 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 wavelet packet 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 wavelet packet 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 IFFT s and FFT s have some problems. The OFDM suffers from ISI (intersymbol interference) This is usually taken care of by using a adding a cyclic prefix greater than the channel length but this may not always be possible. This occurs due to loss of orthogonality due to channel effects.
Time and Frequency Synchronization- The OFDM requires time and frequency synchronization to get a low bit error rate. Carrier Frequency Offset- The offset between the carrier frequency and the frequency 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 leads to 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. For the 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 no decomposition.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 then passed 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
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 FFT s in the OFDM are just replaced by the DWT/IDWT to give the DWT-OFDM The 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:
For an AWGN channel the parameters used are 4 subcarriers, BPSK modulation and bi-orthogonal wavelets. In [4] it is shown that some of the wavelets 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 AWGN channel doesn t 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 much better 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
Fig.10 Performance of DFT-OFDM and DWT-OFDM over a AWGN channel 2. Flat Rayleigh fading channel For 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<25dB) while DFT-OFDM is better for some. For the bi-orthogonal wavelet packets we find that only a few wavelets have worse performance than the single subcarrier OFDM.
Fig 11. Performance of DFT-OFDM and DWT-OFDM over Rayleigh channel The further advantages of the Wavelet OFDM are 1. That it requires lesser overhead as it doesn t require CP 2. It doesn t 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 Wavelet s reduces the overhead thus giving a larger bandwidth. The wavelet packet is much better than the implementing just the wavelet transform as it is more flexible. The bi-orthogonal wavelets though may provide some advantage and better flexibility doesn t perform well for some wavelets considered to the theoretical case. References 1. Communication Systems, 4 th edition, Simon Haykin, John Wiley and Sons,Inc.. 2. C. V. Bouwel, et. al, Wavelet Packet Based Multicarrier Modulation, IEEE Communications and Vehicular Technology, SCVT 200, pp. 131-138, 2000 3. B. G. Nagesh, H. Nikookar, Wavelet Based OFDM for Wireless Channels, IEEE Vehicular Technology Conference, Vol. 1, pp. 688-691, 2001. 4. LI Wiehua, et.al, Bi-orthogonal Wavelet Packet based Multicarrier modulation 5. 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)