1时域采样 a discrete fourier Transform assumes n samples in time and frequency. a By sampling in time, we get a periodic spectrum with the sampling frequency fs.The approximation of a Fourier transform by a df t is reasonable only if the frequency components of X(t are concentrated on a smaller range than the Nyquist frequency f/2
◼ Discrete Fourier Transform assumes N samples in time and frequency. ◼ By sampling in time, we get a periodic spectrum with the sampling frequency fs . The approximation of a Fourier transform by a DFT is reasonable only if the frequency components of x(t) are concentrated on a smaller range than the Nyquist frequency fs /2. 1 时域采样
2频域采样 对x(m)的频谱X(e)进行理想采样后频谱x(e)为: 2)=X(e)6an,(9)=X(e2)∑o(9-k3) x,(e)所对应的时域序列x(n)为: Fa2(_2) Fe"yF|∑a0-2)
2 频域采样 =− = = − k s j j j s X e X e X e k S ( ) ( ) ( ) ( ) ( ) = − = =− − − − − k s j j s F X e F k x n F X e F S ( ) * ( ) ( ) ( ) * ( ) 1 1 1 1 对 x(n) 的频谱 X (e j ) 进行理想采样后频谱 ( ) 为： j s X e ( ) 所对应的时域序列 为： j s X e x (n) s
∑6(n-N 2丌 )=∑2(92-k) Sk=-∞ 2丌 ∑ 2丌 2丌 F∑(0-A2)=0∑0 S 所以 (m)=x(m)*|a∑(n sl=-∞ 2 ∑x(m)*(n-1)=∑ x(n
=− =− − − = − l s s k s F k n l ) 2 ( 1 ( ) 1 =− =− =− − = − = − = s l s s l s s l s s x n n l x n l x n x n n l ) 2 ( 1 ) 2 ( )* ( 1 ) 2 ( 1 ( ) ( )* N s 2 ) 2 ( 2 ( ) =− =− = − − l s k s s N k N F n lN ) ( ) 2 ( 1 s s l s k F n l = − k − =− =− 所以
结论: (1)DFT以2m/N为间隔离散化X(e),所以事实上DFT是 以刚好不产生时域混叠的采样间隔对x(e0)进行的采样。 (2如果对Xe2进行更密的采样,即,=22,则等 效于在时域中取o,N-]范围内的x(n)进行周期延拓, 其中x(n)=,当n=N,N+1…(N-1),如图所示 (3)如果N<M,即对X(e")进行欠采样,则将造成时 域的混叠,如图所示
结论： (1)DFT以 为间隔离散化 ，所以事实上DFT是 以刚好不产生时域混叠的采样间隔对 进行的采样。 (2)如果对 进行更密的采样，即 ，则等 效于在时域中取[ ]范围内的 进行周期延拓， 其中 ,当 ,如图所示。 (3) 如果 ，即对 进行欠采样，则将造成时 域的混叠，如图所示。 2 / N ( ) j X e ( ) j X e ( ) j X e N s s N 2 2 = 0, −1 Ns = , + 1, ( − 1) s x(n) = 0 n N N N N s N ( ) j X e x(n)