In this dissertation we treat the aliasing error in multi-channel sampling. The aliasing error is the difference between a non-bandlimited function and its representation in a series of the Riesz basis of the Paley-Wiener space. It is a well-known fact that sampling functions in multi-channel sampling comprise the Riesz basis for the Paley-Wiener space even though they are not a translation of one as in ordinary sampling. Thus, we define a series representation by this Riesz basis for a non-bandlimited function and the aliasing error in multi-channel sampling in the same way as in the ordinary sampling. The Poisson summation formula plays a key role in deriving the upper bound for the aliasing error. Contrary to the ordinary sampling, we can derive the upper bound for the aliasing error in multi-channel sampling by having as many Poisson summation formulas as transfer functions. Also, we need some conditions on transfer functions in order to use the Poisson summation formula because the formula does not mean the pointwise convergence in general.
We show that the aliasing error in multi-channel sampling is bounded by a constant and a non-bandlimited function's representation in a series of the Riesz basis of the Paley-Wiener space in multi-channel sampling is an approximate representation to the original function in a sense that we can obtain a series representation with as small aliasing error as we want. We also extend this argument to the multi-dimensional case.
이 논문에서는 다중채널에서의 aliasing error 를 다룬다. 이 aliasing error를 다루기 위해서는 Poisson summation formula 가 이용이 되는데 일반적으로 Poisson summation formula 에서는 등호관계가 성립하지 않는다. 이때 transfer function 에 어떤 조건을 주어 aliasing error 를 구하는데 Poisson summation formula 사용 할수있다. 다중채널에서는 채널의 수만큼 Poisson summation formula 를 만들면 이들의 합이 aliasing error 에서 나타나는 series 가 된다. 마찬가지로 주어진 non-bandlimited 함수도 같은 형태의 series 로 바꾸어 두 series 의 차로서 aliasing error 를 나타낸다. 이 에러는 어떤 상수보다 클수 없음을 볼수가 있고 만약 이 함수를 보다 큰 밴드 영역에 해당하는 샘플링 series 로 나타냄으로써 이 에러를 얼마든지 작게 할수 있음을 볼 수가 있다.
