DT transfer function; drawing a Bode plot

Impulse response (discrete time)

The impulse response $h[n]$ of a discrete-time LTI system is the output when the input is the unit impulse $\delta[n]$ and the system is initially at rest:

$$\delta[n] \rightarrow h[n]$$

Because any input can be written as $x[n]=\sum_k x[k]\delta[n-k]$, the output of an LTI system is fully fixed by $h[n]$ through the convolution sum $y[n]=x[n]*h[n]$.

Transfer function (discrete time)

The transfer function is the ratio of the transform of the output to the transform of the input (zero initial conditions). It equals the transform of $h[n]$:

$$H(z)=\frac{Y(z)}{X(z)}=\sum_{n=-\infty}^{\infty}h[n]z^{-n}, \qquad H(e^{j\omega})=\frac{Y(e^{j\omega})}{X(e^{j\omega})}=\sum_{n}h[n]e^{-j\omega n}$$

$H(e^{j\omega})$ (the frequency response) is $H(z)$ evaluated on the unit circle $z=e^{j\omega}$. Example: for $y[n]=0.5y[n-1]+x[n]$, $H(e^{j\omega})=\frac{1}{1-0.5e^{-j\omega}}$ and $h[n]=(0.5)^n u[n]$.

Procedure for drawing a Bode plot

A Bode plot is a pair of graphs: magnitude in dB, $20\log_{10}|H(j\omega)|$, and phase $\angle H(j\omega)$, both against $\log_{10}\omega$.

1. Write $H(j\omega)$ in standard (time-constant) form, e.g. $H(j\omega)=\frac{K(1+j\omega/z_1)}{j\omega(1+j\omega/p_1)}$.
2. Identify the factors: constant $K$, poles/zeros at the origin, simple real poles/zeros, and quadratic factors. Note each corner frequency ($\omega=z_1, p_1,\dots$).
3. Draw the asymptote of each factor:
   • constant: flat line at $20\log K$ dB, phase 0° (or 180° if $K<0$);
   • pole at origin: line of $-20$ dB/decade through 0 dB at $\omega=1$, phase $-90^\circ$;
   • simple pole: 0 dB up to the corner, then $-20$ dB/decade; phase goes from 0° to $-90^\circ$ (−45° at the corner);
   • simple zero: same but $+20$ dB/decade and $+90^\circ$;
   • quadratic pole: $-40$ dB/decade after $\omega_n$, phase 0° to −180°.
4. Add the magnitude asymptotes (dB adds because log of a product is a sum), starting from the lowest frequency; the slope changes by ±20 dB/dec at each corner.
5. Add the phase curves of all factors at several frequencies.
6. Correct the asymptotic plot if needed (about 3 dB at a simple corner; resonance peak for low damping).

For a discrete-time system the same idea is used with $H(e^{j\omega})$ plotted for $0\le\omega\le\pi$ (it is periodic in $2\pi$), usually on a linear frequency axis.

 dB
  |------\            magnitude: flat, then
  |       \  -20 dB/dec   rolls off after corner
  |        \
  +---------+--------> log w
          w_c