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Z Transform Pairs and Properties
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
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\ $
$ [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
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
Parseval's relation $ \int_{-\infty}^{\infty} |x(t)|^2 dt = \int_{-\infty}^{\infty} |X(f)|^2 df $

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Abstract algebra continues the conceptual developments of linear algebra, on an even grander scale.

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