Sinyal dan Sistem Waktu Diskrit

Main ideas & concepts covered

• Discrete-time signals (time signal)

  • A discrete-time signal (x[n]) is defined only at integer time indices (n) (from (-\infty) to (+\infty)).
  • For non-integer times (e.g., decimals/fractions), the value is not defined.
  • Discrete-time signals can be represented by:
    • Tabulation
    • Graphs
    • Functions
  • Signals can be generated by sampling a continuous-time signal.

• Basic discrete-time building blocks

  1. Impulse / sample unit (\delta[n])
     • Value 1 at (n=0)
     • Value 0 for (n\neq 0)
  2. Unit step (u[n])
     • Value 1 for (n \ge 0) (“positive (n)”)
     • Value 0 for (n < 0)
  3. Ramp / rem unit (as stated: “Rem unit”)
     • Output equals (n) (e.g., if (n=1), output (1); if (n=2), output (2))
     • Value 0 for negative indices (as stated)

• Exponential and complex exponential signals

  • Mentioned as part of discrete-time signal forms, including imaginary components (complex exponentials).

---

Classifications of discrete-time signals

Periodic vs. aperiodic

• (x[n]) is periodic if there exists a period (N_0) such that: [ x[n] = x[n+N_0] ]

• If no such period exists → aperiodic.

• Conceptual examples: signals repeating every 3 samples and 6 samples.

Energy and power

• Energy/power can be determined using referenced (but unclear in the transcript) equations.
• Energy signal vs. finite-duration
  • If the signal values are finite/limited, it is called an energy signal (as described).

Even (symmetric) vs odd (asymmetric)

• Even / symmetric if: [ x[n] = x[-n] ]

• Odd if: [ x[n] = -x[-n] ]

• Graphically: mirror symmetry for even signals; opposite shape for odd signals.

---

Basic operations on discrete-time signals

1) Time shifting (time “fighting” in transcript)

• Replace (n) with (n-k) or (n+k) to shift.
• Backward shift / relay shift:
  • (x[n+k]) (stated with “when (k) is positive”)
  • Graph shifts direction depends on sign (wording varies in transcript).
• Advance (forward shift):
  • (x[n-k]) (stated using forms like (x[n+k]) with (k<0))
  • Graph direction also depends on the sign (as described).

2) Folding / mirroring

• Replace (n) with (-n): [ x[-n] ]

• Produces a mirror about the time origin.

3) Addition (summation)

• Add two signals: [ z[n] = x_1[n] + x_2[n] ]

4) Time scaling / multiplication by a constant

• Multiply the signal by a constant (k): [ kx[n] ]

• Transcript note (wording): signal becomes “denser or more sparse” depending on whether the constant is above or below 1 (may be imperfect compared to standard theory).

5) Pointwise multiplication of two signals

• Multiply two signals:
  • Product exists only where both are defined (described as being defined over some interval).

---

Discrete-time systems: input/output relationship

• A discrete-time system relates an input (x[n]) to an output (y[n]).
• General form: [ y[n] = T{x[n]} ] where (T) is the system transformation.

Accumulator system example

• Output accumulated from previous output and current input: [ y[n] = y[n-1] + x[n] ]

• Called relaxed (resting) if initial condition at (n-1) equals 0: [ y[n-1] = 0 ]

---

System interconnections (block-diagram operations)

Common interconnections mentioned:

• Addition
  • (x_1 + x_2)
• Multiplication
  • (x_1[n]\cdot x_2[n])

• Multiply by a constant

  • Example: [ y[n] = ax[n] ]

• Delay element

  • Example: [ y[n] = x[n-1] ]

• Advance element

  • Example: [ y[n] = x[n+1] ]

---

Classifications of discrete-time systems

Static vs. dynamic

• Static: (y[n]) depends only on current input (no past/future dependence).
• Dynamic: (y[n]) depends on past input (and/or future, depending on the system).
• Infinite vs. limited dynamic
  • Limited dynamic: dependence exists only over a finite window.
  • Infinite/unlimited dynamic: dependence extends without bound.

Time-invariant vs. time-variant

• Time-invariant: shifting the input shifts the output in the same way (as suggested by checking that delayed input produces delayed output equivalently).
• Time-variant: the shifting property is not preserved.

Linear vs. non-linear

• Linear if it satisfies the superposition principle: [ T{A_1x_1[n] + A_2x_2[n]} = A_1T{x_1[n]} + A_2T{x_2[n]} ]

Causal vs. non-causal (transcript uses “upset”)

• Causal: output at time (n) depends only on present and past inputs:
  • depends on (x[n], x[n-1], x[n-2], \dots)
  • does not depend on future inputs like (x[n+1], x[n+2], \dots)

Stability

• A system is stable if a bounded input produces a bounded output.
• Transcript later connects stability to impulse response magnitude.

---

Convolution and impulse response criteria

• System output can be computed using convolution of:
  • input (x[n]) with impulse/response (h[n]).
• Convolution properties mentioned (as “police” in transcript):
  • Commutative
  • Identity
  • Distributive
  • Shift
  • Associative

Causality in terms of impulse response

• A system is causal if the impulse response satisfies: [ h[n] = 0 \quad \text{for } n<0 ]

Stability in terms of impulse response

• A system is stable if the impulse response is absolutely summable: [ \sum_n |h[n]| < \infty ]

---

Example methodology: finding impulse response

The video provides an example workflow.

• Given system difference equation (as stated)

  • The impulse response is sought for output (y[n]).
  • Transcript references an equation like: [ y[n] = 0.6\,y[n-1] - 0.08\,y[n-2] ] (formatting unclear in the transcript)

• Process

  1. Collect terms to form the characteristic equation

     • Apply a root (“(\lambda)”) approach using shifting terms, resulting in: [ \lambda^2 + 0.8\lambda - 0.6 = 0 ]

  2. Solve for (\lambda)

     • Roots: [ \lambda = 0.2,\quad \lambda = 0.4 ]

  3. Write the general form of the impulse response

     • Using constants (C_1, C_2): [ h[n] = C_1\left(\frac{1}{5}\right)^n + C_2\left(\frac{2}{5}\right)^n ] (powers shown in the transcript in terms of (\tfrac{1}{5}) and (\tfrac{2}{5}))

  4. Use impulse input

     • Set (x[n] = \delta[n])
     • Response starts at (n=0) (as described).
  5. Evaluate at specific (n) values to solve constants
     • Transcript setup includes:
       • (h[0] = C_1 + C_2 = 1)
       • Another equation using (h[1]), leading to:
         • (C_1 = -1)
         • (C_2 = 2)

  6. Substitute constants into (h[n])

     • Final impulse response (as stated): [ h[n] = -\left(\frac{1}{5}\right)^n u[n] + 2\left(\frac{2}{5}\right)^n u[n] ]

     • Uses (u[n]) to enforce causality (no response for (n<0)).

---

Speakers / sources featured

• Alifia Gina Hanifah — Electrical Engineering, Bandung Institute of Technology (Class of 2021), narrator/presenter.