MIT 6.003 Signals and Systems Self-Study Notes

Source: MIT OCW 6.003 Signals and Systems, Fall 2011, taught by Prof. Dennis Freeman. This is a self-study artifact based on the public OCW materials.

Signals and LTI systems

System abstraction

$$y = T\{x\}$$

Linearity:

$$T\{a x_1 + b x_2\}=aT\{x_1\}+bT\{x_2\}$$

Time invariance:

$$x(t)\to y(t)\Rightarrow x(t-t_0)\to y(t-t_0)$$

$$x[n]\to y[n]\Rightarrow x[n-n_0]\to y[n-n_0]$$

LTI representation

$$y(t)=x(t)*h(t)=\int_{-\infty}^{\infty}x(\tau)h(t-\tau)d\tau$$

$$y[n]=x[n]*h[n]=\sum_{k=-\infty}^{\infty}x[k]h[n-k]$$

For an LTI system, $h$ is enough: know $h$, know the whole system.

Causality and stability

$$\text{causal} \iff h(t)=0\text{ for }t<0,\qquad h[n]=0\text{ for }n<0$$

$$\text{BIBO stable} \iff \int_{-\infty}^{\infty}|h(t)|dt<\infty,\qquad \sum_{n=-\infty}^{\infty}|h[n]|<\infty$$

Complex exponentials and modes

Complex exponentials are eigenfunctions of LTI systems.

$$x(t)=e^{st}\Rightarrow y(t)=H(s)e^{st}$$

$$x[n]=z^n\Rightarrow y[n]=H(z)z^n$$

This is the main reason transforms are useful: LTI convolution becomes scalar multiplication in the transform domain.

Natural modes are set by poles:

$$\text{DT mode: } z_i^n,\qquad \text{CT mode: } e^{s_i t}$$

Stable behavior requires modes to decay:

$$|z_i|<1\quad \text{for causal DT systems}$$

$$\mathrm{Re}(s_i)<0\quad \text{for causal CT systems}$$

Difference equations and Z transform

A discrete-time LTI system often appears as:

$$\sum_{k=0}^{N}a_k y[n-k]=\sum_{m=0}^{M}b_m x[n-m]$$

Taking the Z transform:

$$H(z)=\frac{Y(z)}{X(z)}=\frac{\sum_{m=0}^{M}b_m z^{-m}}{\sum_{k=0}^{N}a_k z^{-k}}$$

Z transform:

$$X(z)=\sum_{n=-\infty}^{\infty}x[n]z^{-n}$$

Important pair:

$$a^n u[n]\leftrightarrow \frac{1}{1-a z^{-1}},\qquad \text{ROC: } |z|>|a|$$

Left-sided version:

$$-a^n u[-n-1]\leftrightarrow \frac{1}{1-a z^{-1}},\qquad \text{ROC: } |z|<|a|$$

Same algebraic transform, different ROC, different time signal. The ROC is not optional.

Causality/stability from poles and ROC

causal right-sided rational system: ROC is outside the outermost pole;
stable system: ROC contains the unit circle;
causal and stable rational system: all poles lie inside the unit circle.

Frequency response:

$$H(e^{j\omega})=H(z)\big|_{z=e^{j\omega}}$$

This is valid when the unit circle is in the ROC.

Differential equations and Laplace transform

A continuous-time LTI system often appears as:

$$\sum_{k=0}^{N}a_k\frac{d^k y(t)}{dt^k}=\sum_{m=0}^{M}b_m\frac{d^m x(t)}{dt^m}$$

Laplace transform:

$$X(s)=\int_{-\infty}^{\infty}x(t)e^{-st}dt$$

System function:

$$H(s)=\frac{Y(s)}{X(s)}=\frac{\sum_{m=0}^{M}b_m s^m}{\sum_{k=0}^{N}a_k s^k}$$

Useful pair:

$$e^{-at}u(t)\leftrightarrow \frac{1}{s+a},\qquad \text{ROC: } \mathrm{Re}(s)>-a$$

Causality/stability from poles and ROC

causal right-sided rational system: ROC is to the right of the rightmost pole;
stable system: ROC contains the $j\omega$ axis;
causal and stable rational system: all poles lie in the open left half-plane.

Frequency response:

$$H(j\omega)=H(s)\big|_{s=j\omega}$$

This is valid when the $j\omega$ axis is in the ROC.

Fourier representations

CT Fourier series

For $x(t+T_0)=x(t)$ and $\omega_0=2\pi/T_0$:

$$x(t)=\sum_{k=-\infty}^{\infty}c_k e^{jk\omega_0t}$$

$$c_k=\frac{1}{T_0}\int_{T_0}x(t)e^{-jk\omega_0t}dt$$

CT Fourier transform

$$X(j\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt$$

$$x(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}X(j\omega)e^{j\omega t}d\omega$$

DT Fourier transform

$$X(e^{j\omega})=\sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}$$

$$x[n]=\frac{1}{2\pi}\int_{-\pi}^{\pi}X(e^{j\omega})e^{j\omega n}d\omega$$

$X(e^{j\omega})$ is periodic with period $2\pi$.

Convolution theorem

$$x*h\leftrightarrow XH$$

Parseval intuition

Energy can be measured in either domain:

$$\int_{-\infty}^{\infty}|x(t)|^2dt=\frac{1}{2\pi}\int_{-\infty}^{\infty}|X(j\omega)|^2d\omega$$

$$\sum_{n=-\infty}^{\infty}|x[n]|^2=\frac{1}{2\pi}\int_{-\pi}^{\pi}|X(e^{j\omega})|^2d\omega$$

Frequency response and Bode intuition

For sinusoidal steady state:

$$x(t)=A\cos(\omega_0t+\phi)$$

$$y(t)=A|H(j\omega_0)|\cos(\omega_0t+\phi+\angle H(j\omega_0))$$

Magnitude controls gain; phase controls delay/advance.

For a first-order low-pass example:

$$H(s)=\frac{1}{1+s/\omega_c}$$

$$|H(j\omega)|=\frac{1}{\sqrt{1+(\omega/\omega_c)^2}},\qquad \angle H(j\omega)=-\tan^{-1}(\omega/\omega_c)$$

At high frequency, a first-order pole contributes approximately $-20$ dB/decade and $-90^\circ$ phase.

Feedback and control

For forward path $G$ and feedback path $K$:

$$H_{\mathrm{closed}}=\frac{G}{1+GK}$$

The characteristic equation is:

$$1+G(s)K(s)=0$$

Key idea: feedback changes poles. Stability is determined by closed-loop poles, not open-loop gain alone.

For negative feedback:

it can reduce sensitivity to plant uncertainty;
it can improve disturbance rejection;
it can also destabilize the system if it moves poles to the wrong side.

Sampling

Sampling interval and sampling frequency:

$$T_s=\frac{1}{f_s},\qquad \Omega_s=\frac{2\pi}{T_s}$$

Impulse-train sampling:

$$x_s(t)=\sum_{n=-\infty}^{\infty}x(nT_s)\delta(t-nT_s)$$

Spectrum after sampling:

$$X_s(j\Omega)=\frac{1}{T_s}\sum_{k=-\infty}^{\infty}X(j(\Omega-k\Omega_s))$$

Nyquist condition for bandlimited signals:

$$\Omega_s>2\Omega_M,\qquad f_s>2B$$

Frequency mapping:

$$\omega=\Omega T_s \pmod{2\pi}$$

Aliasing means different CT frequencies map to the same DT frequency.

Modulation

Multiplication in time shifts spectra in frequency.

$$x(t)\cos(\omega_c t)\leftrightarrow \frac{1}{2}\left[X(j(\omega-\omega_c))+X(j(\omega+\omega_c))\right]$$

This is the basic AM idea: move a baseband signal to a bandpass channel centered at $\omega_c$.

Demodulation multiplies again and then low-pass filters:

$$2x(t)\cos(\omega_c t)\cos(\omega_c t)=x(t)+x(t)\cos(2\omega_c t)$$

The low-pass filter removes the $2\omega_c$ term.

Quantization

For a uniform quantizer with step size $\Delta$:

$$q=x_Q-x,\qquad |q|\le \frac{\Delta}{2}$$

If quantization error is modeled as uniform noise:

$$\sigma_q^2=\frac{\Delta^2}{12}$$

For an ideal $B$-bit quantizer over a fixed range:

$$\Delta \propto 2^{-B}$$

Rough rule: each additional bit improves SQNR by about $6$ dB.

Problem-solving habits

Start by asking whether the system is LTI. If yes, find $h$, $H(z)$, or $H(s)$.
Always keep the ROC with a Z/Laplace transform.
Use poles for modes and stability; use zeros for frequency rejection.
For steady-state sinusoids, use frequency response directly.
For feedback, analyze the closed-loop denominator.
For sampling, draw shifted spectra before doing algebra.