2. Linear Transformations and Matrices

1.1 Continuous - Time & Discrete-time

• Continuous-time signal: $x(t)$, $t \in \mathbb{R}$

• Discrete-time signal: $x[n]$, $n \in \mathbb{Z}$

ex) RC circuit

[회로도: 전압원 $v_s(t)$가 저항 $R$과 커패시터 $C$에 직렬로 연결됨. 출력 전압 $v_c(t)$는 커패시터 양단에서 측정.]

• $Z_R = R$

• $Z_C = \frac{1}{j\omega C}$

Note

• $\omega$ : 입력신호의 주파수 $(rad/s)$

• $f$ : 초당회전 횟수 $(Hz)$

Recall

• In capacitor, $f \rightarrow ?$ … $2\pi f = \omega$.

• $f=0 \rightarrow Z_C = \infty$, open circuit ($V_c = V_s$)

• $f=\infty \rightarrow Z_C = 0$, short circuit

Transfer Function:

\[\frac{V_c(j\omega)}{V_s(j\omega)} = H(j\omega) = \frac{\frac{1}{j\omega C}}{R+\frac{1}{j\omega C}} = \frac{1}{1+j\omega RC}\]

Magnitude:

\[|H(j\omega)| = \frac{1}{\sqrt{1+\omega^2(RC)^2}}\]

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1.1.2 Signal energy & Power

Definition

Continuous-time signal

• $Energy = \int_{t_1}^{t_2}

  x(t)

  ^2 dt$

  • $E_{\infty} = \lim_{T \rightarrow \infty} \int_{-T}^{T}

    x(t)

    ^2 dt$

Discrete-time signal

• $Energy = \sum_{n=n_1}^{n_2}

  x[n]

  ^2$

  • $E_{\infty} = \sum_{n=-\infty}^{\infty}

    x[n]

    ^2$

Definition (Power)

• $Power = \frac{1}{t_2 - t_1} \int_{t_1}^{t_2}

  x(t)

  ^2 dt$

  • $P_{\infty} = \lim_{T \rightarrow \infty} \frac{1}{2T} \int_{-T}^{T}

    x(t)

    ^2 dt$

• Discrete: $P = \frac{1}{2N+1} \sum_{n=-N}^{N}

  x[n]

  ^2$

  • $P_{\infty} = \lim_{N \rightarrow \infty} \frac{1}{2N+1} \sum_{n=-N}^{N}

    x[n]

    ^2$

Remark

• If $P_{\infty} \neq 0 \Rightarrow E_{\infty} = \infty$

• If $E_{\infty} < \infty \Rightarrow P_{\infty} = 0$

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1.2 Transformation of indep. variable

(1) Time Shift

\[x[n] \mapsto x[n - n_0] \quad (n_0 \text{만큼 shift})\]

• if $n_0 > 0$: delay (과거신호)

• if $n_0 < 0$: advance (미래신호)

(2) Time Reversal

\[x[n] \mapsto x[-n]\]

• $n=0$ 대칭 (카세트 reverse)

(3) Time Scaling

• $x[t] \mapsto x[2t]$ : 2배 빠르게 (compressed)

• $x[t] \mapsto x[\frac{1}{2}t]$ : 느리게 (expanded)

• to consider combined transformations… (순서 중요)

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1.2.2 Periodic signals

Definition

If for $\forall t$ s.t.

\[x(t) = x(t+T)\]

$x(t)$ is periodic signal.

$\Rightarrow x(t) = x(t+T)$ 를 만족하는 양수 $T$가 존재한다.

• Fundamental Period: 주기 $T$ 중 최솟값 $T_0$.

Note

• Fundamental frequency: 주파수 중에 가장 작은 주파수 $\omega_0$.

• Is constant signal periodic?

  • $\rightarrow$ Yes, by definition. but no $T_0$ (undefined).

1.2.3 Even & Odd signal

Def)

• Even signal: $x(-t) = x(t)$ (y축 대칭)

• Odd signal: $x(-t) = -x(t)$ (원점 대칭)

$\rightarrow$ We can decompose signal.

\[Ev\{x(t)\} = \frac{x(t) + x(-t)}{2}\] \[Od\{x(t)\} = \frac{x(t) - x(-t)}{2}\] \[\Rightarrow x(t) = Ev\{x(t)\} + Od\{x(t)\}\]

---

1.3 Exponential & Sinusoidal signal

\[x(t) = C e^{at} \quad (C, a \in \mathbb{C})\]

i) If $C, a \in \mathbb{R}$

• $a > 0$: Growing exponential

• $a < 0$: Decaying exponential

ii) If $a$ is pure imaginary number, $a = j\omega_0$

\[x(t) = e^{j\omega_0 t} = \cos(\omega_0 t) + j\sin(\omega_0 t)\]

(Euler’s Formula)

Ex) Find $T$ of periodic $e^{j\omega_0 t}$

\[e^{j\omega_0 t} = e^{j\omega_0 (t+T)}\] \[e^{j\omega_0 t} = e^{j\omega_0 t} \cdot e^{j\omega_0 T}\] \[\Rightarrow e^{j\omega_0 T} = 1\] \[\Rightarrow j\omega_0 T = j2\pi n\] \[\therefore \omega_0 T = 2\pi n \quad \text{or} \quad T = \frac{2\pi}{|\omega_0|} \cdot n\]

• Fundamental period $T_0 = \frac{2\pi}{

  \omega_0

  }$

• 자연계에는 복소수가 없지만 계산 편의를 위해 복소수 signal 사용.

  • $\hookrightarrow$ 실제 측정값은 real part. $Re{x(t)} = \frac{x(t) + x^*(t)}{2}$

Energy & Power of Periodic Signal:

\[E_{period} = \int_{0}^{T_0} |e^{j\omega_0 t}|^2 dt = \int_{0}^{T_0} 1 dt = T_0\] \[P_{period} = \frac{1}{T_0} \cdot E_{period} = 1\]

Harmonically related exponentials

Def) 어떤 양수 주파수 $\omega_0$의 정수배 집합.

\[\phi_k(t) = e^{jk\omega_0 t}, \quad k = 0, \pm 1, \pm 2, \dots\]

Ex 1.5)

\[x(t) = e^{j3t} + e^{j2t}\]

(Change to one cosine function)

\[= e^{j2.5t} (e^{j0.5t} + e^{-j0.5t})\]

(conjugate 관계)

\[= e^{j2.5t} (2 \cos(0.5t))\] \[= 2 e^{j2.5t} \cos(0.5t)\]

Generalized complex-exponential signals

\[C, a \in \mathbb{C}\] \[C = |C| e^{j\theta}, \quad a = r + j\omega_0\] \[x(t) = C \cdot e^{at}\] \[= |C| e^{j\theta} \cdot e^{(r + j\omega_0)t}\] \[= |C| e^{rt} \cdot e^{j\theta} \cdot e^{j\omega_0 t}\] \[= |C| e^{rt} \cdot e^{j(\omega_0 t + \theta)}\] \[= |C| e^{rt} (\cos(\omega_0 t + \theta) + j\sin(\omega_0 t + \theta))\]

[그래프: 진동하며 진폭이 지수적으로 커지는 파형]

• $

  C

  e^{rt}$: 진폭 (Envelope)

• $\omega_0$: 주파수 (Frequency)

• $\theta$: phase offset

---

Discrete-time Signal

\[x[n] = C \cdot \alpha^n\]

(physically meaningful even if $\alpha < 0$)

다음을 관찰하자.

1. $x[n] = \cos(\frac{2\pi}{12} n) \Rightarrow N_0 = 12$

2. $x[n] = \cos(\frac{8\pi}{31} n) \Rightarrow N_0 = 31$

3. $x[n] = \cos(\frac{1}{6} n) \Rightarrow N_0 = \frac{2\pi}{1/6} = 12\pi \dots \notin \mathbb{Z}$ (periodic 하지 않음)

• discrete signal 에서는 $\omega_0 = 2\pi \times (\frac{m}{N})$ 꼴이어야만 periodic 하다.

Periodicity properties of discrete-time complex exponentials

Continuous-time signal 에서는 $\cos(\omega_0 t)$

• 주파수는 $\omega_0$이고, $\sim \infty$까지 의미가 있다.

• $\Rightarrow \omega_0$가 다르면 신호의 의미가 다르다.

• $\Rightarrow$ 음수 주파수는 회전 방향이 반대이다.

Discrete-time signal 의 경우는 $\cos(\omega_0 n)$

• 가장 낮은 주파수: $0$ (or $2\pi, \dots$)

• 가장 높은 주파수: $\pi$ (or $3\pi, \dots$)

• $\pi \sim 2\pi$ 까지는 주파수가 증가해도 주기가 길어지는 효과 (Aliasing).

• 물리적인 의미는 $0$부터 $\pi$까지.

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Discrete-time signal fundamental period

\[e^{j\omega_0 n} = e^{j\omega_0 (n + N_0)}\] \[e^{j\omega_0 n} = e^{j\omega_0 n} \cdot e^{j\omega_0 N_0}\] \[\Rightarrow e^{j\omega_0 N_0} = 1\] \[\Rightarrow \omega_0 N_0 = 2\pi m \quad (m \text{ is integer})\] \[N_0 = \frac{2\pi}{|\omega_0|} \cdot m\]

• 이것이 정수가 되는 $m$을 찾아 계산.

ex) $x[n] = \cos(\frac{8\pi}{31} n)$

\[\omega_0 = \frac{8\pi}{31}\] \[N = \frac{2\pi}{(8\pi/31)} m = \frac{31}{4} m\]

• For $N$ to be integer, let $m=4$.

• $\therefore N_0 = 31$.

Harmonically related periodic exponentials (공통주기 N)

\[\phi_k[n] = e^{jk(\frac{2\pi}{N})n}, \quad k = 0, \pm 1, \dots\]

$\Rightarrow$ 어느 순간 같은 신호가 나올 것이다.

proof)

\[\phi_{k+N}[n] = e^{j(k+N)\frac{2\pi}{N}n}\] \[= e^{jk \frac{2\pi}{N} n} \cdot e^{j N \frac{2\pi}{N} n}\]

이때 $e^{j 2\pi n} = 1$

\[= \phi_k[n] \cdot 1 = \phi_k[n]\]

$\Rightarrow$ $N$개의 서로 다른 harmonically related periodic exponentials 존재.

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1.4 Unit Impulse & Unit step function

(1) Unit Impulse (Discrete)

\[\delta[n] = \begin{cases} 0 & n \neq 0 \\ 1 & n = 0 \end{cases}\]

(2) Unit Step (Discrete)

\[u[n] = \begin{cases} 0 & n < 0 \\ 1 & n \ge 0 \end{cases}\]

Thm (Relationships):

1. $\delta[n] = u[n] - u[n-1]$

2. $u[n] = \sum_{k=0}^{\infty} \delta[n-k] = \sum_{m=-\infty}^{n} \delta[m]$

Properties:

1. $x[n]\delta[n] = x[0]\delta[n]$ ($n=0$ 에서의 sampling)

2. $x[n]\delta[n - n_0] = x[n_0]\delta[n - n_0]$ ($n=n_0$에서의 sampling)

How about continuous-time?

(1) Unit step

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

(Undefined at $t=0$, 이런 함수는 없으니 근사하자)

(2) Unit Impulse

\[\delta(t) = \frac{du(t)}{dt}\]

[그림: 폭 $\Delta$, 높이 $1/\Delta$ 인 사각형 펄스 $\delta_{\Delta}(t)$]

• $\delta(t) \approx \delta_\Delta(t)$

• 넓이가 1로 일정, 적분시 값은 1이다.

• $\delta(t) = \lim_{\Delta \to 0} \delta_\Delta(t)$

• $\int_{-\infty}^{\infty} \delta(t) dt = 1$

What do $u(t)$ and $\delta(t)$ mean?

Note:

\[u(t) = \int_{-\infty}^{t} \delta(\tau) d\tau = \int_{\infty}^{0} \delta(t-\sigma) (-d\sigma) \quad (\text{subst } \sigma = t-\tau)\] \[u(t) = \int_{0}^{\infty} \delta(t-\sigma) d\sigma\]

Properties:

1. $x(t)\delta(t) = x(0)\delta(t)$ ($t=0$ 에서의 신호만 남음)

2. $x(t)\delta(t - t_0) = x(t_0)\delta(t - t_0)$ ($t=t_0$ 에서의 sampling)

3. Sifting Property:

   \[\int_{-\infty}^{\infty} x(\tau) \delta(t - \tau) d\tau = x(t)\]

   (어차피 $\tau=t$ 일 때만 값을 가진다)

   \[= x(t) \int_{-\infty}^{\infty} \delta(t-\tau) d\tau = x(t)\]

Doublet (Derivative of Impulse):

\[\delta'(t) := \frac{d\delta(t)}{dt}\]

By convolution property, $\int_{-\infty}^{\infty} x(\tau) \delta(t - \tau) d\tau = x(t)$

Let $\tau’ = t - \tau$:

\[\frac{d}{dt} \int_{-\infty}^{\infty} x(t - \tau') \delta(\tau') d\tau'\] \[= [x(t-\tau')\delta(\tau')]_{-\infty}^{\infty} - \int_{-\infty}^{\infty} x(t-\tau') \delta'(\tau') d\tau'\]

(Integration by parts)

\[= \int_{-\infty}^{\infty} x(t-\tau') \delta'(\tau') d\tau'\]

(디렉 델타가 1번 미분)

\[\frac{dx(t)}{dt} = \int_{-\infty}^{\infty} x(t-\tau') \delta'(\tau') d\tau'\]

Generalize:

\[\frac{d^n x(t)}{dt^n} = \int_{-\infty}^{\infty} x(t - \tau') \delta^{(n)}(\tau') d\tau'\]

: 자기자신보다 다른 signal 과 결합할 때 의미가 있다.

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Q. How can I prove $\frac{d^{2}x(t)}{dt^{2}} = \int_{-\infty}^{\infty} x(t-\tau’) \delta^{(2)}(\tau’) d\tau’$ ?

1.5 Continuous-Time and Discrete-Time systems

• Continuous: $x(t) \to \boxed{\text{System}} \to y(t)$

• Discrete: $x[n] \to \boxed{\text{System}} \to y[n]$

Ex 1.11 Simulation

Model: $\dot{y}(t) + ay(t) = bx(t)$

Approximate derivative: $\dot{y}(t) \approx \frac{y(t) - y(t-h)}{h}$

\[\frac{y(t) - y(t-h)}{h} + a y(t-h) = b x(t)\] \[y(t) = (1 - ha) y(t-h) + hb x(t)\]

Convert to discrete ($t \to n$, $t-h \to n-1$):

\[y[n] = (1 - ha) y[n-1] + hb x[n]\]

• $y(t-h) \approx y[n-1]$ : 한 sample 이전의 값. 미소 시간 전만큼의 값.

• 근사 가능!

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1.6 Basic System Properties

(1) Memory

• Memoryless system:

  • Ex: $y[n] = 2x[n] - x^2[n]$

  • Ex: $y(t) = Re(x(t))$ 등. 해당시간의 input에만 영향을 받음.

• System with Memory:

  • Ex: $y[n] = \sum_{k=-\infty}^{n} x[k]$

  • Ex: $y(t) = x(t) + x(t+1)$ 등. 과거나 미래 입력을 고려.

  • Ex: $y(t) = \frac{1}{C} \int_{-\infty}^{t} i(\tau) d\tau$

(2) Invertibility

• Invertible system:

  \[x[n] \to \boxed{S} \to y[n] \to \boxed{S^{-1}} \to x[n]\]

  If distinct inputs lead to distinct outputs.

• Non-invertible system:

  • Ex: $y[n] = 0$

  • Ex: $y(t) = x^2(t)$ 등.

(3) Causality

• Causal System: 출력이 입력의 현재와 과거 값에 의해서 결정된다. (non-anticipative).

• Non-causal System: 미래값에 의해 결정.

Note

• All memoryless systems are causal.

• Almost all real-life continuous systems are causal.

• Discrete-time systems do not have to be causal. (Just wait. 모든 sample을 미리 받은 후 실행).

Ex) $y[n] = \frac{1}{2M+1} \sum_{k=-M}^{M} x[n-k]$ (Moving Average)

(4) Stability

[그림: (상단) 진자 운동 - 안정, (하단) 역진자 - 불안정]

“BIBO”: Bounded Input, Bounded Output.

유한한 입력 $\rightarrow$ 유한한 출력.

Ex: $y(t) = e^{x(t)}$

i) $\forall

x(t)

< B \Rightarrow

y(t)

=

e^{x(t)}

\le e^{

x(t)

} < e^B < \infty$

ii) $\exists t$ s.t. $

x(t)

< B \dots$ (내용 끊김, 불안정 조건 반례에 대한 언급으로 추정됨)

(5) Time Invariance

\[x(t) \to \boxed{S} \to y(t)\] \[x(t - t_0) \to \boxed{S} \to y(t - t_0) \quad \text{for } \forall t_0\]

$\therefore t_0$ delay $\rightarrow$ 출력도 $t_0$ delay : 언제 실험하든 같은 결과.

Diagram:

1. $x(t) \xrightarrow{\text{Shift}} x(t-t_0) \xrightarrow{\text{Sys}} Y_1$

2. $x(t) \xrightarrow{\text{Sys}} y(t) \xrightarrow{\text{Shift}} y(t-t_0)$

\[T \cdot S_{t_0} = S_{t_0} \cdot T \quad \text{일 때 Time-invariant.}\]

(6) Linearity

1. Additivity: $x_1[n] + x_2[n] \to y_1[n] + y_2[n]$

2. Homogeneity: $a x_1[n] \to a y_1[n]$

보이는 법: $x[n]$ 대신에 $a x_1[n] + b x_2[n]$ 넣기