信号与系统课件--§4.5 傅里叶变换的性质.ppt

1、4.5 傅里叶变换的性质,线性 奇偶性 对称性 尺度变换 时移特性,频移特性 卷积定理 时域微分和积分 频域微分和积分 相关定理,一线性性质(Linear Property),If f1(t) F1(j)， f2(t) F2(j) then,a f1(t) + b f2(t) a F1(j) + b F2(j) ,Proof: F a f1(t) + b f2(t),= a F1(j) + b F2(j) ,Example,二奇偶虚实性(Parity),If f (t) is real function, and,f (t) F(j)=|F(j)|ej() = R()+jX(),then,R(

2、)= R()， X()= X()， |F(j)|= |F(j)|， ()= ()， f (t) F(j) = F*(j) If f (t)= f (t) then X()=0， F(j) = R() If f (t)= f (t) then R()=0， F(j) = jX(),Proof,三、对称性(Symmetrical Property),If f (t) F(j) then,Proof:,（1）,in (1) t ，t then,（2）,in (2) - then, F(j t) 2f () end,F( jt ) 2f (),Example,练习,四、尺度变换性质(Scaling T

3、ransform Property),If f (t) F(j) then,where “a” is a nonzero real constant.,Proof,Also,letting a = -1,f (- t ) F( -j),Example-1,意义,五、时移特性(Timeshifting Property),If f (t) F(j) then,where “t0” is real constant.,Proof: F f (t t0 ) ,Example 1,Example 2,Example 3,六、频移性质(Frequency Shifting Property),If f

4、(t) F(j) then,Proof:,where “0” is real constant.,F e j0t f(t),= F j(-0) end,For example 1,f(t) = ej3t F(j) = ?,Ans: 1 2() ej3t 1 2(-3),Example 2,七、卷积性质(Convolution Property),Convolution in time domain：,If f1(t) F1(j)， f2(t) F2(j) Then f1(t)*f2(t) F1(j)F2(j),Convolution in frequency domain：,If f1(t)

5、F1(j)， f2(t) F2(j),Then f1(t) f2(t) F1(j)*F2(j),Proof,Example,八、时域的微分和积分(Differentiation and Integration in time domain),If f (t) F(j) then,Proof:,f(n)(t) = (n)(t)*f(t) (j)n F(j) f(-1)(t)= (t)*f(t) ,Example 1,Example 2,已知f (t) F1(j) f (t) F (j)=?,九、频域的微分和积分(Differentiation and Integration in frequency domain),If f (t) F(j) then,(jt)n f (t) F(n)(j),where,Example 1,Example 2,十、相关定理(Correlation theorem),If f1(t) F1(j) ，f2(t) F2(j) ，f