The discrete Fourier transform, Hadamard transform (or Walsh transform), and convolution transform all play important roles in signal processing and coding practice in several engineering disciplines. In this chapter, we introduce these transforms for periodic sequences over a finite field . The discrete Fourier transform yields the trace representation of a periodic sequence. The number of nonzero Fourier spectral coefficients is equal to the linear span of the sequence. These results are included in Sections 6.1 6.3. The trace representation of a periodic sequence over is a function from to . We discuss this one-to-one correspondence in Section 6.4. Furthermore, the Hadamard transform and convolution transform of sequences can be defined in terms of their trace representations, which are introduced in Sections 6.5. Consequently, the correlation between two sequences is transferred to correlation between their trace representations. This is presented in Section 6.6. In Section 6.7, we show some basic properties of the Hadamard transform and convolution transform. Thus, the Fourier transform serves as a bridge for a connection between sequences and functions. It is worth pointing out that all newly discovered binary sequences with 2-level autocorrelation functions were proved in terms of their Hadamard transforms. Therefore, the contents introduced in this chapter are fundamental tools for the design of sequences with special properties. The last section features the discrete Fourier transform and the Hadamard transform in their matrix representations.