傅里葉變換的性質(zhì)課件

1、4.5 傅里葉變換的性質(zhì) 線性 奇偶性 對稱性 尺度變換 時移特性 頻移特性 卷積定理 時域微分和積分 頻域微分和積分 相關定理一、線性性質(zhì)(Linearity Property)If f1(t) F1(j)， f2(t) F2(j)thena 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、奇偶虛實性(Parity)If f (t) is real function, and f (t) F(j)=|F(j)|ej() = R()+jX()then R()= R(

2、)， 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、對稱性(Symmetry/Symmetrical Property)If f (t) F(j) thenProof:（1）in (1) t ，t then （2）in (2) - then F(j t) 2f () endF( jt ) 2f ()Example練習四、尺度變換性質(zhì)(Scaling Transform

3、 Property)If f (t) F(j) then where “a” is a nonzero real constant.Also,letting a = -1,f (- t ) F( -j) Example-1意義五、時移特性(Time-shifting Property)If f (t) F(j) thenwhere “t0” is real constant.Proof: F f (t t0 ) Example 1Example 2Example 3六、頻移性質(zhì)(Frequency Shifting Property)If f (t) F(j) thenProof:where

4、“0” is real constant.F e j0t f(t)= F j(-0) endFor example 1f(t) = ej3t F(j) = ?Ans: 1 2() ej3t 1 2(-3)Example 2七、卷積性質(zhì)(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) F1(j)， f2(t) F2(j)Then f1(t) f2(t) F1(j

5、)*F2(j)ProofExample八、時域的微分和積分(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 1Example 2已知f (t) F1(j) f (t) F (j)=?九、頻域的微分和積分(Differentiation and Integration in frequency domain)If f (t) F(j) then (jt)n f (t) F(n)(j) whereExample 1Example 2十、相關定理(Correlation theorem)If f1(t) F1(j) ，f2(t) F2(j) ，f(