On signal processing with sampling errors: a worst case analysis
The starting point of this paper is a communication system by which a sequence \((s_ k)_{k\in {\mathbb{Z}}}\) of signals is modulated into a continuous signal of the form \[ y(t) = T\sum^{\infty}_{j=- \infty}s_ j h(t-jT),\quad t\in {\mathbb{R}}. \] The signals \(s_ k\) are assumed to be identically distributed real random variables with probability space (A,\({\mathfrak A},\mu)\) such that \[ \sum^{\infty}_{j=- \infty}| s_ j(a)| <\infty \quad for\quad \mu -almost\quad all\quad a\in A, \] E[s\({}_ k(\cdot)]=0\), \(E[s_ k(\cdot)^ 2]=1\) for all \(k\in {\mathbb{Z}}\), and \(E[s_ j(\cdot)s_ k(\cdot)]=0\) for all j,k\(\in {\mathbb{Z}}\) with \(j\neq k\) where \(E[S]=\int_{A}s(a)d\mu (a)\). The impulse response h is chosen as a real valued function is \(L^ 2({\mathbb{R}})\) such that its Fourier transform vanishes outside some interval [- \(\Omega\),\(\Omega\) ], \(\Omega >0\). If the modulated signal \((y(\cdot,t))_{t\in {\mathbb{R}}}\) is sampled in the form (y(\(\cdot,k\cdot T))\), \(k\in {\mathbb{Z}}\), then exact sampling is possible, i.e. \(y(a,k\cdot T)=s_ k(a)\) for all \(k\in {\mathbb{Z}}\) and \(\mu\)-almost all \(a\in A\), if, for instance, h is chosen as \(h_ n(t)=1/T\) for \(t=0\), \(h_ n(t)=(1/T)\sin (n\pi t/T)/(n\pi t/T)\) for \(t\neq 0\) and some \(n\in {\mathbb{N}}\) (in which case we have that \(\Omega =\Omega_ n=[- (n/2)\omega_ 0\), \((n/2)\omega_ 0]\) with \(\omega_ 0=2\pi /T).\) The paper is now concerned with the case in which y(\(\cdot\), \(k\cdot T+v(t))\) is sampled instead of y(\(\cdot\), \(k\cdot T)\) where \(v=v(t)\) is a random variable with probability space (\({\mathcal T}=[-\tau,\tau],{\mathfrak B},p)\) for some \(\tau\in [0,T/2]\), \({\mathfrak B}\) being the sigma algebra of Borel sets in \({\mathcal T}\) and p being a Borel measure on \({\mathfrak B}\) with p(\({\mathcal T})=1\). By v(t) a random sampling error is expressed whose measure space is assumed to be (\({\mathcal T},{\mathfrak B})\) and whose probability distribution is then given by \(G_ p(t)=p\{t\in {\mathcal T}| v(\hat t)\leq t\}\). If p is given, one can try to find h such that the mean square error \[ E[(y(\cdot,k\cdot T+v(t))-s_ k(\cdot))^ 2]=\int_{A}[y(a,k\cdot T+t)-s_ k(a))^ 2dG_ p(t) \] (which, under the above assumptions, is independent of k) becomes minimal. A worst case analysis then consists of the determination of p such that the corresponding minimal mean square error becomes maximal. This problem is discussed in general and is completely solved for the case that h is chosen in the space of functions spanned by \(h_ 1\) and by \(h_ 1\), \(h_ 2\), respectively, with \(h_ n\) as defined above. Concerning the case where h is chosen in the span of \(h_ 1,...,h_ N\) for any \(N\in {\mathbb{N}}^ a \)conjecture is made (which in the meantime could be proved).
scientific article; zbMATH DE number 4014605