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## Continuous and Discrete Wavelet Analysis

These examples aim to show the difference between the discrete wavelet transform (DWT) and the continuous wavelet transform (CWT).

When is Continuous Analysis More Appropriate than Discrete Analysis?

To answer this, consider the related questions: Do you need to know all values of a continuous decomposition to reconstruct the signal exactly? Can you perform nonredundant analysis? When the energy of the signal is finite, not all values of a decomposition are needed to exactly reconstruct the original signal, provided that you are using a wavelet that satisfies some admissibility condition. Usual wavelets satisfy this condition. In this case, a continuous-time signal is characterized by the knowledge of the discrete transform. In such cases, discrete analysis is sufficient and continuous analysis is redundant.

Example 1: Fractal Signal

When a signal is recorded in continuous time or on a very fine time grid, both analyses are possible. Which should be used? It depends; each one has its own advantages.

```load vonkoch;
lv = 510;
signal = vonkoch(1:lv);
plot(signal,'r');
set(gca,'XLim',[1 length(signal)])
title('Analyzed signal');
xlabel('Time (or Space)')
ylabel('Amplitude')
```

Discrete Wavelet Transform (DWT)

Discrete analysis ensures space-saving coding and is sufficient for exact reconstruction. Perform discrete wavelet transform at level 5 by sym2:

```lev   = 5;
wname = 'sym2';
nbcol = 64;
[c,l] = wavedec(signal,lev,wname);
```

Expand the discrete wavelet coefficients for visualization

```len = length(signal);
cfd = zeros(lev,len);
for k = 1:lev
d = detcoef(c,l,k);
d = d(:)';
d = d(ones(1,2^k),:);
cfd(k,:) = wkeep1(d(:)',len);
end
cfd =  cfd(:);
I = find(abs(cfd)<sqrt(eps));
cfd(I) = zeros(size(I));
cfd = reshape(cfd,lev,len);
cfd = wcodemat(cfd,nbcol,'row');
```

Plot the discrete coefficients.

```colormap(pink(nbcol));
image(cfd);
tics = 1:lev;
labs = int2str((1:lev)');
set(gca,...
'YTickLabelMode','manual','YDir','normal', ...
'Box','On','YTick',tics,'YTickLabel',labs);
title('Discrete Wavelet Transform, Absolute Coefficients.');
xlabel('Time (or Space)')
ylabel('Level');
```

Continuous Wavelet Transform (CWT)

Continuous analysis is often easier to interpret, since its redundancy tends to reinforce the traits and makes all information more visible. This is especially true of very subtle information. Thus, the analysis gains in "readability" and in ease of interpretation what it loses in terms of saving space.

```scales = (1:32); % Levels 1 to 5 correspond to scales 2, 4, 8, 16 and 32.
cwt(signal,scales,wname,'plot');
colormap(pink(nbcol));
```

Example 2: Frequency Break

Let's do the same comparison on a second signal. The purpose of this example is to show how analysis by wavelets can detect the exact instant when a signal changes. The discontinuous signal consists of a slow sine wave abruptly followed by a medium sine wave.

```load freqbrk;
signal = freqbrk;
```

Perform the discrete wavelet transform (DWT) at level 5 using db1.

```lev   = 5;
wname = 'db1';
nbcol = 64;
[c,l] = wavedec(signal,lev,wname);
```

Expand discrete wavelet coefficients for plot.

```len = length(signal);
cfd = zeros(lev,len);
for k = 1:lev
d = detcoef(c,l,k);
d = d(:)';
d = d(ones(1,2^k),:);
cfd(k,:) = wkeep1(d(:)',len);
end
cfd =  cfd(:);
I = find(abs(cfd)<sqrt(eps));
cfd(I) = zeros(size(I));
cfd    = reshape(cfd,lev,len);
cfd = wcodemat(cfd,nbcol,'row');
```

Perform the continuous wavelet transform (CWT) and visualize results

```set(subplot(3,1,1),'XTick',[]);
plot(signal,'r');
title('Analyzed signal.');
set(gca,'XLim',[1 length(signal)])
subplot(3,1,2);
colormap(cool(128));
image(cfd);
tics = 1:lev;
labs = int2str(tics');
set(gca,...
'YTickLabelMode','manual','YDir','normal', ...
'Box','On','YTick',tics,'YTickLabel',labs ...
);
title('Discrete Transform, absolute coefficients.');
ylabel('Level');
set(subplot(3,1,2),'XTick',[]);
subplot(3,1,3);
scales = (1:32);
cwt(signal,scales,wname,'plot');
colormap(cool(128));
tt = get(gca,'YTickLabel');
[r,c] = size(tt);
yl = char(32*ones(r,c));
for k = 1:3:r , yl(k,:) = tt(k,:); end
set(gca,'YTickLabel',yl);
```

Here is a noteworthy example of an important advantage of wavelet analysis over Fourier. If the same signal had been analyzed by the Fourier transform, we would not have been able to detect the instant when the signal's frequency changed, whereas it is clearly observable here.

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