7.1 Fundamentals of Signal and Systems

7.1 Fundamentals of Signal and Systems

1. Signal Classification

Signals are functions that convey information about the state or behavior of a physical system. They are classified based on different characteristics.

1.1 Continuous-time vs. Discrete-time Signals

1. Continuous-time (CT) Signals:

   • Defined for all values of time t in a given interval.

   • Represented as $x(t)$.

   • Examples: Audio signals, temperature readings, voltage in an analog circuit.

2. Discrete-time (DT) Signals:

   • Defined only at specific, discrete instants of time.

   • Usually obtained by sampling a CT signal at regular intervals.

   • Represented as $x[n]$, where n is an integer index.

   • Examples: Digital audio, monthly sales data, pixel values in an image.

1.2 Periodic vs. Aperiodic Signals

1. Periodic Signals:

   • Repeat their pattern over a fixed time interval.

   • CT: $x(t) = x(t + T_0)$ for all t, where $T_0$ is the fundamental period.

   • DT: $x[n] = x[n + N]$ for all n, where N is the fundamental period (integer).

   • Examples: Sine wave, square wave, sawtooth wave.

2. Aperiodic (Non-periodic) Signals:

   • Do not repeat their pattern over time.

   • Examples: Speech signal, unit step function, most real-world random signals.

1.3 Energy vs. Power Signals

1. Energy Signal:

   • Has finite total energy and zero average power.

   • CT Energy: $E = \int_{-\infty}^{\infty} |x(t)|^2 dt < \infty$

   • DT Energy: $E = \sum_{n=-\infty}^{\infty} |x[n]|^2 < \infty$

   • Examples: Finite-duration pulses, decaying exponentials.

2. Power Signal:

   • Has finite average power and infinite total energy.

   • CT Power: $P = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T} |x(t)|^2 dt < \infty$

   • DT Power: $P = \lim_{N \to \infty} \frac{1}{2N+1} \sum_{n=-N}^{N} |x[n]|^2 < \infty$

   • Examples: Periodic signals, unit step function, random signals.

1.4 Even vs. Odd Signals

1. Even Signals:

   • Symmetric about the vertical axis (time=0).

   • CT: $x(t) = x(-t)$

   • DT: $x[n] = x[-n]$

   • Examples: Cosine function, $x(t) = t^2$.

2. Odd Signals:

   • Antisymmetric about the origin.

   • CT: $x(t) = -x(-t)$

   • DT: $x[n] = -x[-n]$

   • Examples: Sine function, $x(t) = t^3$.

3. Decomposition: Any signal can be expressed as the sum of its even and odd parts:

   • CT: $x(t) = x_e(t) + x_o(t)$ where $x_e(t) = \frac{x(t) + x(-t)}{2}$ and $x_o(t) = \frac{x(t) - x(-t)}{2}$

   • DT: Similar decomposition applies.

1.5 Orthogonal Signals

1. Definition: Two signals are orthogonal if their inner product is zero.

2. CT Orthogonality: $\int_{-\infty}^{\infty} x_1(t) x_2^*(t) dt = 0$

3. DT Orthogonality: $\sum_{n=-\infty}^{\infty} x_1[n] x_2^*[n] = 0$

4. Significance: Orthogonal signals don't interfere with each other; basis for Fourier series and transforms.

1.6 Causal/Anticausal/Noncausal Signals

1. Causal Signals:

   • Zero for all negative time.

   • CT: $x(t) = 0$ for $t < 0$

   • DT: $x[n] = 0$ for $n < 0$

   • Examples: Unit step function, real-world signals that start at t=0.

2. Anticausal Signals:

   • Zero for all positive time.

   • CT: $x(t) = 0$ for $t > 0$

   • DT: $x[n] = 0$ for $n > 0$

3. Noncausal Signals:

   • Non-zero for both positive and negative time.

   • Examples: Most theoretical signals, even functions.

2. Signal Transformations

These operations modify signals in the time domain.

2.1 Time Shifting

1. Operation: $y(t) = x(t - t_0)$ or $y[n] = x[n - n_0]$

2. Effect:

   • $t_0 > 0$: Shift right (delay).

   • $t_0 < 0$: Shift left (advance).

3. Example: $x(t-2)$ shifts the signal 2 units to the right.

2.2 Time Scaling

1. Operation: $y(t) = x(at)$

2. Effect:

   • $|a| > 1$: Compression (signal speeds up).

   • $0 < |a| < 1$: Expansion (signal slows down).

   • $a < 0$: Includes time reversal.

3. Example: $x(2t)$ compresses the signal by factor of 2.

2.3 Time Reversal

1. Operation: $y(t) = x(-t)$ or $y[n] = x[-n]$

2. Effect: Flips the signal about the vertical axis.

3. Example: Mirror image of the original signal.

2.4 Combined Transformations

• General form: $y(t) = x(at - b)$

• Order matters: Usually perform scaling, then shifting on the scaled signal.

• Correct approach: $y(t) = x(a(t - \frac{b}{a}))$

3. Standard Signals

3.1 Unit Impulse (Delta Function)

1. Continuous-time ($\delta(t)$):

   • Definition: $\delta(t) = 0$ for $t \neq 0$, $\int_{-\infty}^{\infty} \delta(t) dt = 1$

   • Properties:

     • Sifting: $\int_{-\infty}^{\infty} x(t)\delta(t-t_0) dt = x(t_0)$

     • Sampling: $x(t)\delta(t-t_0) = x(t_0)\delta(t-t_0)$

2. Discrete-time ($\delta[n]$):

   • Definition: $\delta[n] = \begin{cases} 1 & n = 0 \\ 0 & n \neq 0 \end{cases}$

   • Properties: $x[n]\delta[n-n_0] = x[n_0]\delta[n-n_0]$

3.2 Unit Step Function

1. Continuous-time ($u(t)$):

   • $u(t) = \begin{cases} 1 & t > 0 \\ 0 & t < 0 \end{cases}$

   • Relation with impulse: $\frac{du(t)}{dt} = \delta(t)$

2. Discrete-time ($u[n]$):

   • $u[n] = \begin{cases} 1 & n \geq 0 \\ 0 & n < 0 \end{cases}$

   • Relation: $u[n] = \sum_{k=-\infty}^{n} \delta[k]$

3.3 Unit Ramp Function

1. Continuous-time ($r(t)$):

   • $r(t) = t u(t) = \begin{cases} t & t \geq 0 \\ 0 & t < 0 \end{cases}$

   • Relations: $\frac{dr(t)}{dt} = u(t)$, $\frac{d^2r(t)}{dt^2} = \delta(t)$

2. Discrete-time ($r[n]$):

   • $r[n] = n u[n] = \begin{cases} n & n \geq 0 \\ 0 & n < 0 \end{cases}$

3.4 Exponential Signals

1. Continuous-time:

   • Real: $x(t) = e^{at}$ (growing if a>0, decaying if a<0)

   • Complex: $x(t) = e^{j\omega_0 t} = \cos(\omega_0 t) + j\sin(\omega_0 t)$

2. Discrete-time:

   • Real: $x[n] = a^n$

   • Complex: $x[n] = e^{j\Omega_0 n} = \cos(\Omega_0 n) + j\sin(\Omega_0 n)$

3.5 Signum Function

1. Continuous-time ($sgn(t)$):

   • $sgn(t) = \begin{cases} 1 & t > 0 \\ 0 & t = 0 \\ -1 & t < 0 \end{cases}$

   • Relation: $sgn(t) = 2u(t) - 1$

2. Discrete-time: Similar definition applies.

4. System Properties

A system transforms an input signal into an output signal: $y(t) = T\{x(t)\}$

4.1 Linearity

1. Definition: A system is linear if it satisfies both:

   • Additivity: $T\{x_1(t) + x_2(t)\} = T\{x_1(t)\} + T\{x_2(t)\}$

   • Homogeneity (Scaling): $T\{ax(t)\} = aT\{x(t)\}$

2. Combined (Superposition): $T\{a_1x_1(t) + a_2x_2(t)\} = a_1T\{x_1(t)\} + a_2T\{x_2(t)\}$

3. Examples: Circuits with resistors, capacitors, inductors (linear elements).

4.2 Time-Invariance

1. Definition: A system is time-invariant if a time shift in the input causes the same time shift in the output:

   • If $y(t) = T\{x(t)\}$, then $T\{x(t-t_0)\} = y(t-t_0)$ for any $t_0$.

2. Test: Apply input $x(t)$ → get output $y(t)$. Then apply input $x(t-t_0)$ → output should be $y(t-t_0)$.

3. Examples: Systems with constant parameters (not changing with time).

4.3 Causality

1. Definition: The output at any time depends only on present and past inputs, not future inputs.

2. Mathematically: For CT: $y(t_0)$ depends on $x(t)$ for $t \leq t_0$. For DT: $y[n_0]$ depends on $x[n]$ for $n \leq n_0$.

3. Physical significance: Real-time systems must be causal.

4. Examples: Real-time filters, control systems.

4.4 Stability (BIBO Stability)

1. Definition: A system is BIBO (Bounded Input Bounded Output) stable if every bounded input produces a bounded output.

2. Mathematically: If $|x(t)| \leq M_x < \infty$ for all t, then $|y(t)| \leq M_y < \infty$ for all t.

3. Test: Check if impulse response is absolutely integrable (CT) or absolutely summable (DT).

4. Importance: Ensures system doesn't "blow up" with finite inputs.

4.5 Memory (Dynamical Property)

1. Memoryless (Static) System: Output at time t depends only on input at time t.

   • Example: $y(t) = 2x(t)$, $y(t) = x^2(t)$

2. System with Memory (Dynamic): Output depends on past/future inputs.

   • Example: $y(t) = x(t-1)$ (delay), $y(t) = \int_{-\infty}^{t} x(\tau) d\tau$

5. Linear Time-Invariant (LTI) Systems

5.1 Definition and Importance

1. LTI Systems: Systems that are both Linear and Time-Invariant.

2. Importance:

   • Can be completely characterized by their impulse response.

   • Output is convolution of input with impulse response.

   • Fourier/Laplace transforms provide powerful analysis tools.

5.2 Impulse Response

1. Definition: The output of an LTI system when the input is a unit impulse $\delta(t)$ or $\delta[n]$.

2. Notation: $h(t)$ for CT, $h[n]$ for DT.

3. Significance: Completely characterizes the LTI system.

6. Convolution

6.1 Continuous-time Convolution Integral

1. Definition: For LTI system with impulse response $h(t)$ and input $x(t)$: $y(t) = x(t) * h(t) = \int_{-\infty}^{\infty} x(\tau) h(t-\tau) d\tau$

2. Properties:

   • Commutative: $x(t) * h(t) = h(t) * x(t)$

   • Associative: $[x(t) * h_1(t)] * h_2(t) = x(t) * [h_1(t) * h_2(t)]$

   • Distributive: $x(t) * [h_1(t) + h_2(t)] = x(t) * h_1(t) + x(t) * h_2(t)$

3. Graphical Interpretation: Flip, shift, multiply, integrate.

6.2 Discrete-time Convolution Sum

1. Definition: For LTI system with impulse response $h[n]$ and input $x[n]$: $y[n] = x[n] * h[n] = \sum_{k=-\infty}^{\infty} x[k] h[n-k]$

2. Properties: Similar to CT convolution (commutative, associative, distributive).

3. Length considerations: If $x[n]$ has length L and $h[n]$ has length M, then $y[n]$ has length L+M-1.

6.3 Convolution with Special Signals

1. With impulse: $x(t) * \delta(t-t_0) = x(t-t_0)$

2. With step: $x(t) * u(t) = \int_{-\infty}^{t} x(\tau) d\tau$

3. Two impulses: $\delta(t-t_1) * \delta(t-t_2) = \delta(t-(t_1+t_2))$

6.4 Steps for Computing Convolution

1. For CT:

   • Express signals in terms of $\tau$.

   • Flip $h(\tau)$ to get $h(-\tau)$.

   • Shift by t to get $h(t-\tau)$.

   • Multiply $x(\tau)$ and $h(t-\tau)$.

   • Integrate over $\tau$ where product is non-zero.

2. For DT:

   • Similar steps using summation instead of integration.

   • Consider all possible overlaps between flipped/shifted $h[n-k]$ and $x[k]$.

7. Important Relationships and Formulas

7.1 Signal Energy and Power Formulas

1. CT Signal Energy: $E = \int_{-\infty}^{\infty} |x(t)|^2 dt$

2. CT Signal Power: $P = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T} |x(t)|^2 dt$

3. DT Signal Energy: $E = \sum_{n=-\infty}^{\infty} |x[n]|^2$

4. DT Signal Power: $P = \lim_{N \to \infty} \frac{1}{2N+1} \sum_{n=-N}^{N} |x[n]|^2$

7.2 Even-Odd Decomposition

1. CT Signal: $x(t) = x_e(t) + x_o(t)$ where $x_e(t) = \frac{x(t) + x(-t)}{2}$, $x_o(t) = \frac{x(t) - x(-t)}{2}$

2. DT Signal: Similar decomposition with $x[n]$ and $x[-n]$

7.3 System Property Tests

1. Linearity Test: Check if $T\{a_1x_1 + a_2x_2\} = a_1T\{x_1\} + a_2T\{x_2\}$

2. Time-invariance Test: Check if $T\{x(t-t_0)\} = y(t-t_0)$

3. Causality Test: Check if output at $t_0$ depends only on $x(t)$ for $t \leq t_0$

4. Stability Test: Check if $\int_{-\infty}^{\infty} |h(t)| dt < \infty$ (CT) or $\sum_{n=-\infty}^{\infty} |h[n]| < \infty$ (DT)

7.4 Convolution Properties

1. Identity: $x(t) * \delta(t) = x(t)$

2. Time shift: $x(t) * \delta(t-t_0) = x(t-t_0)$

3. Differentiation: $\frac{d}{dt}[x(t) * h(t)] = \frac{dx(t)}{dt} * h(t) = x(t) * \frac{dh(t)}{dt}$

4. Integration: $\int_{-\infty}^{t} [x(\tau) * h(\tau)] d\tau = [\int_{-\infty}^{t} x(\tau) d\tau] * h(t) = x(t) * [\int_{-\infty}^{t} h(\tau) d\tau]$