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1、3.1 离散傅里叶变换的定义 3.2 离散傅里叶变换的基本性质3.3 频率域采样3.4 DFT的应用举例第3章 离散傅里叶变换(DFT)Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.3.
2、1 离散傅里叶变换的定义 3.1.1 DFT的定义设x(n)是一个长度为M的有限长序列，则定义x(n) 的N点离散傅里叶变换为X(k)的离散傅里叶逆变换为Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose P
3、ty Ltd.式中 ,N称为DFT变换区间长度NM，通常称(3.1.1) 式和(3.1.2)式为离散傅里叶变换对。例 3.1.1 x(n)=R4(n) ，求x(n)的8点DFT 设变换区间N=8， 则Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyr
4、ight 2004-2011 Aspose Pty Ltd.3.1.2 DFT和Z变换的关系设序列x(n)的长度为N，其Z变换和DFT分别为：比较上面二式可得关系式Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Asp
5、ose Pty Ltd.图 3.1.1 X(k)与X(e j)的关系 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.3.1.3 DFT的隐含周期性前面定义的DFT变换对中，x(n)与X(
6、k)均为有限长序列，但由于WknN的周期性，使(3.1.1)式和(3.1.2)式中X(k)隐含周期性，且周期均为N。对任意整数m，总有均为整数 所以(3.1.1)式中， X(k)满足同理可证明(3.1.2)式中 x(n+mN)=x(n)Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 As
7、pose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.实际上，任何周期为N的周期序列 都可以看作长度为N的有限长序列x(n)的周期延拓序列，而x(n)则是 的一个周期，即为了以后叙述方便， 将(3.1.5)式用如下形式表示： Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyri
8、ght 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.图 3.1.2 有限长序列及其周期延拓 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 A
9、spose Pty Ltd.式中x(n)N表示x(n)以N为周期的周期延拓序列， (n)N表示n对N求余，即如果n=MN+n1， 0n1N-1， M为整数，则 (n)N=n1例如， 则有所得结果附合图2.1.2所示的周期延拓规律。 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspo
10、se Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.如果x(n)的长度为N，且 (n)=x(n)N，则可写出 (n)的离散傅里叶级数为(3.1.8) (3.1.9) 式中 (3.1.10)Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose
11、 Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.3.2 离散傅里叶变换的基本性质3.2.1 线性性质如果x1(n)和x2(n)是两个有限长序列，长度分别为N1和N2。 y(n)=ax1(n)+bx2(n)式中a、b为常数，即N=maxN1, N2，则y(n)N点DFT为Y(k)=DFTy(n)=aX1(k)+bX2(k), 0kN-1(3.2.1)其中X1(k)和X2(k)分别为x1(n)和x2(n)的N点DFT。 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3
12、.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.3.2.2 循环移位性质1. 序列的循环移位设x(n)为有限长序列，长度为N，则x(n)的循环移位定义为y(n)=x(n+m)NRN(N) (3.2.2)Evaluation only.Evaluation only. Created with Aspose.Slides for
13、 .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.图 3.2.1 循环移位过程示意图 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose
14、.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.2. 时域循环移位定理设x(n)是长度为N的有限长序列，y(n)为x(n)的循环移位，即y(n)=x(n+m)NRN(n)则Y(k)=DFTy(n)=WN-kmX(k) 其中X(k)=DFTx(n), 0kN-1。 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Clie
15、nt Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.3. 频域循环移位定理如果X(k)=DFTx(n), 0kN-1Y(k)=X(k+l)NRN(k)则 y(n)=IDFTY(k)=WNnlx(n) (3.2.4)Evaluation only.Evaluation only. Created with Aspose.Slides for .N
16、ET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.3.2.3 循环卷积定理有限长序列x1(n)和x2(n)，长度分别为N1和N2， N=max N1, N2 。x1(n)和x2(n)的N点DFT分别为：X1(k)=DFTx1(n)X2(k)=DFTx2(n)如果 X(k)=X1(k)X2(k) 则(3.2.5) Evaluation only.Evaluation only. Creat
