已知信号x(t)=3cos8πt，从t=0开始以采样周期Ts=0.1s采样得到序列x(n),分别取采样点数N=5，N=8，计算X(k)=DFT[x(n)]

当N=5时，采样点为n=0,1,2,3,4，对应的采样时间为t=0,0.1,0.2,0.3,0.4

则序列x(n)=[3,0,-3,0,3]

计算DFT：

X(k)=∑x(n)exp(-j2πkn/N)

当k=0时，X(0)=∑x(n)=3+0-3+0+3=3

当k=1时，X(1)=∑x(n)exp(-j2πn/N)=3exp(-j2π/5)+0+(-3)exp(-j4π/5)+0+3exp(-j6π/5)=-3-3j

当k=2时，X(2)=∑x(n)exp(-j4πn/5)=3exp(-j4π/5)+0+(-3)exp(-j8π/5)+0+3exp(-j12π/5)=-1.83697e-15-6.12323e-16j

当k=3时，X(3)=∑x(n)exp(-j6πn/5)=3exp(-j6π/5)+0+(-3)exp(-j12π/5)+0+3exp(-j18π/5)=3+3j

当k=4时，X(4)=∑x(n)exp(-j8πn/5)=3exp(-j8π/5)+0+(-3)exp(-j16π/5)+0+3exp(-j24π/5)=1.83697e-15+6.12323e-16j

因此，当N=5时，DFT结果为X(k)=[3,-3-3j,-1.83697e-15-6.12323e-16j,3+3j,1.83697e-15+6.12323e-16j]

当N=8时，采样点为n=0,1,2,3,4,5,6,7，对应的采样时间为t=0,0.1,0.2,0.3,0.4,0.5,0.6,0.7

则序列x(n)=[3,2.12,-0.88,-3,-2.12,0.88,3,2.12]

计算DFT：

X(k)=∑x(n)exp(-j2πkn/N)

当k=0时，X(0)=∑x(n)=0

当k=1时，X(1)=∑x(n)exp(-j2πn/8)=0.693+1.792j

当k=2时，X(2)=∑x(n)exp(-j4πn/8)=-5.88e-16+1.41421j

当k=3时，X(3)=∑x(n)exp(-j6πn/8)=-0.693+1.792j

当k=4时，X(4)=∑x(n)exp(-j8πn/8)=0

当k=5时，X(5)=∑x(n)exp(-j10πn/8)=0.693-1.792j

当k=6时，X(6)=∑x(n)exp(-j12πn/8)=1.18e-15-1.41421j

当k=7时，X(7)=∑x(n)exp(-j14πn/8)=-0.693-1.792j

因此，当N=8时，DFT结果为X(k)=[0,0.693+1.792j,-5.88e-16+1.41421j,-0.693+1.792j,0,0.693-1.792j,1.18e-15-1.41421j,-0.693-1.792j]