Transcription of Fourier Transform Theorems Addition Theorem Shift …
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Fourier Transform Theorems Addition Theorem Shift …
Fourier Transform Theorems Addition Theorem Shift Theorem Convolution Theorem Similarity Theorem Rayleigh s Theorem Differentiation TheoremAddition TheoremF{f+g}=F+GProof:F{f+g}(s) =Z [f(t) +g(t)]e j2 stdt=Z f(t)e j2 stdt+Z g(t)e j2 stdt=F(s) +G(s) Shift TheoremF{f(t t0)}(s) =e j2 st0F(s)Proof:F{f(t t0)}(s) =Z f(t t0)e j2 stdtMultiplying the byej2 st0e j2 st0=1 yields:F{f(t t0)}(s)=Z f(t t0)e j2 stej2 st0e j2 st0dt=e j2 st0Z f(t t0)e j2 s(t t0) t0anddu=dtyields:F{f(t t0)}(s) =e j2 st0Z f(u)e j2 sudu=e j2 st0F(s). Shift Theorem ExampleF{sin(2 (t+1/4))}(s)=ej2 s4F{sin(2 t)}=ej s2 j2[ (s+1) (s 1)]=j2[ej s2 (s+1) ej s2 (s 1)]=j2[ej ( 1)2 (s+1) ej (+1)2 (s 1)]=j2[ j (s+1) j (s 1)]=12[ (s+1) + (s 1)]=F{cos(2 s)} Shift Theorem (variation)F 1{F(s s0)}(t) =ej2 s0tf(t)Proof:F 1{F(s s0)}(t) =Z F(s s0)ej2 stdsMultiplying the byej2 s0te j2 s0t=1 yields:F 1{F(s s0)}(t)=Z F(s s0)ej2 stej2 s0te j2 s0tds=ej2 s0tZ F(s s0)ej2 (s s0) s0anddu=dsyields:F 1{F(s s0)}(t) =ej2 s0tZ F(u)ej2 utdu=ej2 s0tf(t).
Shift Theorem F {f(t −t0)}(s) =e−j2πst0F(s) Proof: F {f(t −t0)}(s) = Z ∞ −∞ f(t −t0)e−j2πstdt Multiplying the r.h.s. by ej2πst0e−j2πst0 =1 yields: F {f(t −t0)}(s) Z ∞ −∞ f(t −t0)e−j2πstej2πst0e−j2πst0dt = e−j2πst0 Z ∞ −∞ f(t −t0)e−j2πs(t−t0)dt. Substituting u =t −t0 and du =dt yields: F {f(t −t0)}(s) = e−j2πst0 Z ∞
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