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Fountain codes are suitable for data transmission over binary erasure channels. When such codes are applied in live broadcast applications, due to the limited bandwidth and delay constraints, it is often convenient to treat the input media stream as a concatenation of short data blocks and to encode each block individually. However, the use of a small block size may result in poor coding efficiency. In this paper, we introduce an approach to enhance the coding efficiency by encoding over progressively more source bits. Our approach can be considered as a generalization of the expanding window fountain code concept. In addition, we provide an error analysis based on state generating functions. The state generating function is useful both as a performance evaluation tool and as a design criterion for the generalized expanding window codes.