To fix the instability of the Shannon representation, we assume that
the signal is slightly more bandlimited than before
f ^(ω)=0for|ω|≥π-δ,δ>0,f ^(ω)=0for|ω|≥π-δ,δ>0,(1)
and instead of using
χ [-π,π] χ [-π,π] , we multiply by another
function
g ^(ω)g ^(ω) which is very similar in form to the
characteristic function, but decays at its boundaries in a smoother
fashion (i.e. it has more derivatives). A candidate function
g ^g ^ is sketched in
Figure 1.
Now, it is a property of the Fourier transform that an increased
smoothness in one domain translates into a faster decay in the
other. Thus, we can fix our instability problem, by choosing
g ^g ^ so that g ^g ^ is smooth and g ^(ω)=1g ^(ω)=1, |ω|≤π-δ|ω|≤π-δ
and g ^=0g ^=0, |ω|>π|ω|>π. By choosing the smoothness of gg suitably large, we can, for any given m≥1m≥1, choose gg to satisfy
|g(t)|≤C (|t|+1) m |g(t)|≤C (|t|+1) m (2)
for some constant
C>0C>0.
Using such a g ^g ^, we can rewrite ((Reference))
as
f ^(ω)=F(ω)g ^(ω)=1 2π∑ n∈Z f(n)e -inω g ^(ω).f ^(ω)=F(ω)g ^(ω)=1 2π∑ n∈Z f(n)e -inω g ^(ω).(3)
Thus, we have the new representation
f(t)=∑ n∈Z f(n)g(t-n),f(t)=∑ n∈Z f(n)g(t-n),(4)
where we gain stability from our additional assumption that the
signal is bandlimited on
[-π-δ,π-δ][-π-δ,π-δ].
Does this assumption really hurt? No, not really because if our
signal is really bandlimited to [-π,π][-π,π] and not
[-π-δ,π-δ][-π-δ,π-δ], we can always take a slightly larger
bandwidth, say [-λπ,λπ][-λπ,λπ] where λλ is a
little larger than one, and carry out the same analysis as above.
Doing so, would only mean slightly oversampling the signal (small
cost).
Recall that in the end we want to convert analog
signals into bit streams. Thus far, we have the two representations
f(t)=∑ n∈Z f(n) sinc (π(t-n)),f(t)=∑ n∈Z fn λg(λt-n).f(t)=∑ n∈Z f(n) sinc (π(t-n)),f(t)=∑ n∈Z fn λg(λt-n).(5)
Shannon's Theorem tells us that if
f∈B A f∈B A , we should
sample
ff at the Nyquist rate
AA (which is twice the support of
f ^f ^) and then take the binary
representation of the samples. Our more stable representation says
to slightly oversample
ff and then convert to a binary
representation. Both representations offer perfect reconstruction,
although in the more stable representation, one is straddled with
the additional task of choosing an appropriate
λλ.
In practical situations, we shall be interested in approximating ff
on an interval [-T,T][-T,T] for some T>0T>0 and not for all time.
Questions we still want to answer include
- How many bits do we need to represent ff in B A=1 B A=1 on some interval [-T,T][-T,T] in the norm L ∞ [-T,T]L ∞ [-T,T]?
- Using this methodology, what is the optimal way of encoding?
- How is the optimal encoding implemented?
Towards this end, we define
B A :={f∈L 2 (R):|f ^(ω)|=0,|ω|≥Aπ}.B A :={f∈L 2 (R):|f ^(ω)|=0,|ω|≥Aπ}.(6)
Then for any
f∈B A f∈B A , we can write
f=∑ n fn A·sincπ(At-n).f=∑ n fn A·sincπ(At-n).(7)
In other words, samples at 0,
±1 A±1 A,
±2 A,⋯±2 A,⋯ are sufficient to reconstruct
ff. Recall also
that
sinc(x)=sin(x) xsinc(x)=sin(x) x decays poorly
(leading to numerical instability). We can overcome this problem by
slight over-sampling. Say we over-sample by a factor
λ>1λ>1.
Then, we can write
f=∑fn λAg λ (λAt-n).f=∑fn λAg λ (λAt-n).(8)
Hence we need samples at 0, ±1 λA±1 λA, ±2 λA±2 λA, etc.
What is the advantage? Sampling more often than necessary buys us stability because we now have a choice
for g λ (·)g λ (·).
If we choose g λ (·)g λ (·) infinitely differentiable whose Fourier transform looks
as shown in Figure 2 we can obtain
|g λ (t)|≤c λ,k (1+|t|) k ,k=1,2,...|g λ (t)|≤c λ,k (1+|t|) k ,k=1,2,...(9)
and therefore
g λ (·)g λ (·) decays very fast. In other words,
a sample's influence is felt only locally. Note however, that
over-sampling generates basis functions that are redundant (linearly
dependent), unlike the integer translates of the
sinc(·)sinc(·) function.
If we restrict our reconstruction to tt in the interval [-T,T][-T,T], we will only need
samples only from [-cT,cT][-cT,cT], for c>1c>1 (see Figure 3),
because the distant samples will have little effect on the reconstruction in
[-T,T][-T,T].
