Z Transform Pairs and Properties | |
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Definition Z Transform and its Inverse | |
Single-side Z Transform | $ X(z)=\mathcal{L}(x[n])=\sum^{\infty}_{n=0}x[n]z^{-n} $ |
Double-side Z Transform | $ X(z)=\mathcal{L}(x[n])=\sum_{n=-\infty}^{\infty}x[n]z^{-n} $ |
Inverse Z Transform | $ x[n]=\mathcal{L}^{-1}(X(z))=\frac{1}{2\pi j}\oint_{c}X(z)z^{n-1}dz $ |
Z Transform Pairs | |||||
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Signal $ x[n] $ | Transform $ X(z) $ | ROC | |||
Unit impulse signal | $ \delta[n]\ $ | $ 1\ $ | $ All\ z\ $ | ||
Unit step signal | $ u[n]\ $ | $ \frac{1}{1-z^{-1}} $ | $ |z| > 1\ $ | ||
$ -u[-n-1]\ $ | $ \frac{1}{1-z^{-1}} $ | $ |z| < 1\ $ | |||
Shifted unit impulse signal | $ \delta[n-m]\ $ | $ z^{-m}\ $ | $ All\ z, except\ 0\ (if\ m>0)\ or\ \infty \ (if\ m<0)\ $ | ||
$ \alpha^{n}u[n]\ $ | $ \frac{1}{1-\alpha z^{-1}} $ | $ |z| > | \alpha |\ $ | |||
$ -\alpha^{n}u[-n-1]\ $ | $ \frac{1}{1-\alpha z^{-1}} $ | $ |z| < | \alpha |\ $ | |||
$ n\alpha^{n}u[n]\ $ | $ \frac{\alpha z^{-1}}{(1-\alpha z^{-1})^{2}} $ | $ |z| > | \alpha |\ $ | |||
$ -n\alpha^{n}u[-n-1]\ $ | $ \frac{\alpha z^{-1}}{(1-\alpha z^{-1})^{2}} $ | $ |z| < | \alpha |\ $ | |||
Single-side cosine signal | $ [\cos{\omega_{0}n}]u[n]\ $ | $ \frac{1-[\cos{\omega_{0}}]z^{-1}}{1-[2\cos{\omega_{0}}]z^{-1}+z^{-2}} $ | $ |z| > 1\ $ | ||
Single-side sine signal | $ [\sin{\omega_{0}n}]u[n]\ $ | $ \frac{1-[\sin{\omega_{0}}]z^{-1}}{1-[2\cos{\omega_{0}}]z^{-1}+z^{-2}} $ | $ |z| > 1\ $ | ||
$ [r^{n}\cos{\omega_{0}n}]u[n]\ $ | $ \frac{1-[r\cos{\omega_{0}}]z^{-1}}{1-[2r\cos{\omega_{0}}]z^{-1}+r^{2}z^{-2}} $ | $ |z| > r\ $ | |||
Single-side cosine signal | $ [r^{n}\sin{\omega_{0}n}]u[n]\ $ | $ \frac{1-[r\sin{\omega_{0}}]z^{-1}}{1-[2r\cos{\omega_{0}}]z^{-1}+r^{2}z^{-2}} $ | $ |z| > r\ $ |
CT Fourier Transform Properties | |||
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x(t) | $ \longrightarrow $ | $ X(f) $ | |
multiplication property | $ x(t)y(t) \ $ | $ X(f)*Y(f) =\int_{-\infty}^{\infty} X(\theta)Y(f-\theta)d\theta $ | |
convolution property | $ x(t)*y(t) \! $ | $ X(f)Y(f) \! $ | |
time reversal | $ \ x(-t) $ | $ \ X(-f) $ |
Other CT Fourier Transform Properties | |
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Parseval's relation | $ \int_{-\infty}^{\infty} |x(t)|^2 dt = \int_{-\infty}^{\infty} |X(f)|^2 df $ |