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Where the proposed scheme varies from traditional OFDM systems is in the compressive unit where compression, modulation and parallel multiplexing all take place together.

The data from the previous baseband encoding block has a 2-dimensional DWT applied to it. Since the M-QAM encoded data is in the form of complex numbers the DWT process is applied to both the real and imaginary components of d(m)

n . The result of this wavelet transform process is still a parallel stream of

complex numbers D(m) n .

The general equation for a 1-dimensional DWT process is shown in equations (5.1) and (5.2). This process consists of a multi-resolution decomposition of an original series of data samples (x[n]).

DW Tx(j, k) = cj,k = X n x[n]h∗ j[n − 2jk] (5.1) bj,k = X n x[n]g∗ J[n − 2Jk] (5.2)

where J is the desired resolution of the decomposition, j = 1, 2, ..., J; k is the discrete time shift; hj[n − 2j] is the discrete wavelet function; gJ[n − 2J] is

the scaling sequence; cj,k are the wavelet coefficients, and bj,k are the scaling

coefficients.

(5.1) and (5.2) the parallel data stream (Dn(m)) out of the DWT block for the

proposed system can be derived. In one-dimension the wavelet and scaling coefficients obtained from the mth _{OFDM block will be:}

c(m)_j,k =X n ℜ(d(m)[n])h∗ j[n − 2jk] + i X n ℑ(d(m)[n])h∗ j[n − 2jk] (5.3) b(m)_j,k =X n ℜ(d(m)[n])g∗ J[n − 2Jk] + i X n ℑ(d(m)[n])g∗ J[n − 2Jk] (5.4)

This process is then repeated in the horizontal direction to obtain the 2- dimensional DWT of the data. The result of this entire process is the creation of an approximation subband, which is simply a downsampled version of the original data, as well as additional subbands, these being the horizontal, vertical and diagonal detail coefficients. The number of these additional subbands is dependant on the desired resolution (J).

In the proposed system the output of this transform block consists of com- plex numbers representing these subband coefficients of the signal. To achieve compression low-pass filtering is applied to the coefficients to eliminate the high frequency coefficients. The low-pass filtering ensures only the approximation and horizontal subband components are retained for transmission.

In the case of image compression this wavelet subband compression process is traditionally performed on the raw image data (with pixel values consisting of integers between 0 and 255) in a separate compressive block before performing

an OFDM multiplexing operation on the data. A potential issue that needed to be considered when altering this traditional method and performing the wavelet subband compression on QAM encoded symbols was whether this method would degrade the reconstructed data more than when using the traditional compres- sion method. Figures 5.3 and 5.4 show the reconstruction of the Lena image after a wavelet subband compression operation is performed on raw pixel values (figure 5.3) and QAM encoded data (figure 5.4). These results are purely from considering compression and reconstruction using the two different methods, at this point no OFDM transmission or channel effects have been introduced.

Figure 5.3: Original image (left) and reconstructed image (right) after image is wavelet subband compressed

Figure 5.4: Original image (left) and reconstructed image (right) after image is both baseband encoded and wavelet subband compressed

From these images it is clear that there is very little visible difference between the reconstructed images for both these methods. Since subbands have been removed in both cases there is some loss of data and hence errors occurring. To quantify this data loss the three error measures described in chapter 4 were used, these are: relative root-mean squared error (RMSE), structural similarity (SSIM) and peak signal-to-noise ratio (PSNR).

From these error measure algorithms a quantitative error analysis can be performed on the two compression techniques. Average RMSE and SSIM values were calculated over the 69 images, for wavelet subband compression on raw pixel values using a Daubechies order 6 wavelet filter (example shown in figure 5.3) an average RMSE value of 0.075 was achieved, this corresponded to an average structural similarity of 83.2% and a PSNR value of 32.7dB, compared to the wavelet compression of 256-QAM encoded symbols (example shown in

figure 5.4) where an average RMSE of 0.081, an average structural similarity of 81% and a PSNR value of 32.7dB was achieved.

These error measurements show a negligible increase in error for the pro- posed compression method. Looking deeper into the proposed method, the relationship between pixel values and the QAM values each of these pixels is mapped to is considered more closely.

Figure 5.5 shows two maps of the in-phase and quadrature QAM components that each pixel in the Lena image is mapped to. These two maps were produced by treating each amplitude component on the in-phase and quadrature axes of the QAM constellation as pixel values to produce a map of values between −15 and 15. These values were then mapped to colours varying from dark blue for the lowest value to dark red for the highest value. It’s very interesting to note the strong visual correlation between the in-phase QAM component and the original Lena image, even the quadrature component has some weak visual correlation to the original image.

Figure 5.5: In-phase (left) and quadrature (right) components of the 256-QAM mod- ulated Lena image

This relationship further confirms that the proposed method of modulating and wavelet compressing an image in one block is a viable method since the spatial frequencies of an image and the correlation between neighbouring pixels remains consistent even after modulation mapping. This is very important since the structure of a wavelet decomposition depends heavily on these properties of an image. The fact there is a very small increase in the RMSE for the two methods can be weighed up against the potential for reduction in computational complexity.