This book by the Fourier monoply, Ronald Bracewell, is absolutely the best book on the Fourier transform. The concepts are very clearly illustrated with easily understood plain English. The whole complicated theory has been made so simple in this book. And more impressively, the whole book only employes very little complicated mathematics and still keep the theory complete and mathematically rigorous.
When reading other books on Fourier theories, I always find the authors rely too much on the mathematical derivations to develope the theory and they themselves seem to not comprehend the essence of the theory. While in this book, Bracewell made every concept intuitively sensible.
I will recommend every one who wants to learn Fourier theory to start from this book rather than any other books, which are usually rather confusing and not self-contained. You can get a 10 times deeper understanding on the whole theory with 10 times less difficulty.