Translation Followed by Compression

Second, rescale the given domain of the integer-shifted basis elements $\phi(t-\ell)$ . This rescaling yields a different basis for a different, but related, vector space. For the Shannon basis this is achieved by again using Eq.(2.79), but by first setting

$$\varepsilon=2^{-k}\varepsilon_0 \quad k=integer$$

before letting $\varepsilon_0=2\pi$ . The result is

$$\sqrt{2^{-k}}\,\phi(2^{-k}t-\ell)=\sqrt{2^{-k}}\,\frac{\sin\pi(2^{-k}t-\ell)} {\pi(2^{-k}t-\ell)}~.$$ (282)

For each integer $k$ these functions form an orthonormal basis for the space of those band-limited functions, whose Fourier domain is restricted to the frequency band $[-\pi2^{-k},\pi2^{-k}]$ . The orthonormality is guaranted by the fact that these functions are derived from the set of orthonormal wave packets $P_{0\ell}(t)$ . The vector space

$$span\left\{\sqrt{2^{-k}}\,\phi(2^{-k}t-\ell):\, \ell=0,\pm1,\cdots\right\} \equiv \mathbf V_k$$ (283)

is called the $k$ th resolution space. For fixed $k$ these basis elements form the $k$ th resolution Shannon basis, more simply the $k$ th Shannon basis. They have the common phase space shape

$$\Delta t = 2^k$$

$$\Delta\omega = \frac{2\pi}{2^k}$$

$$\Delta t\Delta \omega = 2\pi~.$$

These shapes are illustrated in Figure 2.20 for the vector spaces $\mathbf V_{k},\,k=-1,0,1$ . Relative to the phase space cells of $\mathbf V_0$ , $k>0$ implies that the phase space cells get dilated in the time domain and compressed in the frequency domain in order to comply with $\Delta t\Delta \omega=2\pi$ .

Also note that increasing $k$ designates increasing roughness, i.e, lower resolution. Thus increasing resolutions are labelled by decreasing integers. This labelling, which at first sight is backward, highlights the fact that the low resolution features of a signal are generally more significant than those of high resolution.