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| − | '''''Conjugation Property''' | + | '''''Conjugation Property and Conjugate Symmetry''' |
The conjugation property states that if the <math>\mathcal{F}</math> of x(t) will be equal to X(jw) | The conjugation property states that if the <math>\mathcal{F}</math> of x(t) will be equal to X(jw) | ||
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<math>X^* (jw)=[\int\limits_{-\infty}^{\infty}x(t)e^{(-\jmath wt)}dt]^*</math> | <math>X^* (jw)=[\int\limits_{-\infty}^{\infty}x(t)e^{(-\jmath wt)}dt]^*</math> | ||
| − | <math>X^* (jw)=\int\limits_{-\infty}^{\infty}x(t)e^{(-\jmath wt)}dt | + | <math>X^* (jw)=\int\limits_{-\infty}^{\infty}x^* (t)e^{(\jmath wt)}dt</math> |
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| + | Replacing w with -w, | ||
| + | <math>X^* (-jw)=\int\limits_{-\infty}^{\infty}x^* (t)e^{(-\jmath wt)}dt</math> | ||
| + | (The right hand side of this equation is the Fourier transform analysis equation for x*(t)) | ||
| + | |||
| + | The conjugation property shows that if x(t) is real, then X(jw) has conjugate symmetry | ||
| + | <math> X(-\jmath w) = X^* (\jmath w)</math> | ||
| + | where [x(t) real] | ||
Latest revision as of 04:37, 9 July 2009
Conjugation Property and Conjugate Symmetry
The conjugation property states that if the $ \mathcal{F} $ of x(t) will be equal to X(jw) then, the $ \mathcal{F} $ of x*(t) will be equal to X*(-jw) This property follows from the evaluation of the complex conjugate of $ X^* (jw)=[\int\limits_{-\infty}^{\infty}x(t)e^{(-\jmath wt)}dt]^* $ $ X^* (jw)=\int\limits_{-\infty}^{\infty}x^* (t)e^{(\jmath wt)}dt $ Replacing w with -w, $ X^* (-jw)=\int\limits_{-\infty}^{\infty}x^* (t)e^{(-\jmath wt)}dt $ (The right hand side of this equation is the Fourier transform analysis equation for x*(t)) The conjugation property shows that if x(t) is real, then X(jw) has conjugate symmetry $ X(-\jmath w) = X^* (\jmath w) $ where [x(t) real]
