8.1 Representation of Periodic Sequence: the discrete Fourier series 2丌 2丌 2丌 k+mN)n 2mn Due to the periodicity of the complex exponential we only need n exponentials for discrete time Fourier series 刘=∑x[k]e j(2I/N)kn k=0 ◆ No need f[n]=i2Xkk]e7 8.1 Representation of Periodic Sequence: the Discrete Fourier Series ◆Due to the periodicity of the complex exponential we only need N exponentials for discrete time Fourier series (2 ) 1 0 1 / [ ] N k j N kn x n X k e N − = = ◆No need 1 (2 / ) [ ] j N kn k x n X k e N = ( ) ( ) 2 2 2 2 j k n j kn j kn N mN N N j mn e e e e + = =