信号与系统教案第4章.ppt

4.5 傅里叶变换的性质 4.5 傅里叶变换的性质 4.5 傅里叶变换的性质 4.5 傅里叶变换的性质 4.5 傅里叶变换的性质 4.6 周期信号傅里叶变换 4.7 LTI系统的频域分析 4.7 LTI系统的频域分析 4.5 傅里叶变换的性质 For example F(jω) = ? Ans: f (t) = f1(t) – g2(t) f1(t) = 1 ←→ 2πδ(ω) g2(t) ←→ 2Sa(ω) ∴ F(jω) = 2πδ(ω) - 2Sa(ω) ‖ - 4.5 傅里叶变换的性质 二、时移性质(Timeshifting Property) If f (t) ←→F(jω) then where “t0” is real constant. Proof: F [ f (t – t0 ) ] 4.5 傅里叶变换的性质 For example F(jω) = ? Ans: f1(t) = g6(t - 5) , f2(t) = g2(t - 5) g6(t - 5) ←→ g2(t - 5) ←→ ∴ F(jω) = ‖ + 4.5 傅里叶变换的性质 三、对称性质(Symmetrical Property) If f (t) ←→F(jω) then Proof: （1） in (1) t →ω，ω→t then （2） in (2) ω → -ω then ∴ F(j t) ←→ 2πf (–ω) end F( jt ) ←→ 2πf (–ω) 4.5 傅里叶变换的性质 For example ←→ F(jω) = ? Ans: if α=1, ∴ * if F(jω) = ? 4.5 傅里叶变换的性质 四、频移性质(Frequency Shifting Property) If f (t) ←→F(jω) then Proof: where “ω0” is real constant. F [e jω0t f(t)] = F[ j(ω-ω0)] end For example 1 f(t) = ej3t ←→ F(jω) = ? Ans: 1 ←→ 2πδ(ω) ej3t ×1←→ 2πδ(ω-3) 4.5 傅里叶变换的性质 For example 2 f(t) = cosω0t ←→ F(jω) = ? Ans: F(jω) = π[δ(ω+ω0)+ δ(ω-ω0)] For example 3 Given that f(t) ←→ F(jω) The modulated signal f(t) cosω0t ←→ ? 4.5 傅里叶变换的性质 五、尺度变换性质(Scaling Transform Property) If f (t) ←→F(jω) then where “a” is a nonzero real constant. Proof: F [ f (a t ) ] = For a 0 , F [ f (a t ) ] for a 0 , F [ f (a t ) ] That is , f (a t ) ←→ Also,letting a = -1, f (- t ) ←→ F( -jω) 演示 4.5 傅里叶变换的性质 For example 1 Given that f (t)←→F( jω), find f (at – b) ←→ ? Ans: f (t – b)←→ e -jωb F( jω) f (at – b) ←→ or f (at) ←→ f (at – b) = For example 2 f(t) = ←→ F(jω) = ? Ans: Using symmetry, using scaling property with a = -1, so that, 六、卷积性质(Convolution Property) Convolution in time domain： If f1(t) ←→F1(jω)， f2(t) ←→F2(jω) Then f1(t)*f2(t) ←→F1(jω)F2(jω) Convolution in frequency domain： If f1(t) ←→F1(jω)， f2(t) ←→F2(jω) Then f1(t) f2(t) ←→ F1(jω)*F2(jω) 4.5 傅里叶变换的性质 Proof: F [ f1