[디지털신호처리] Discrete Time Signal Processing Chapter 2 내용 정리 (1)

책 : Discrete Time Signal Processing 3rd Edition - Alan V. Oppenheim, Ronald W. Schafer

 

impulse function = delta function = unit impulse (δ[n])

   δ[n] = 1 (n=0)

unit step은 impulse function의 합 (-무한대 ~ n) (u[n])

   u[n] = 1 (n>=0)

discrete signal : sequence of numbers

sample : individual signal value at an index

real exponential : x[n] = Aa^n

sinusoidal : x[n] = cos(w0n+Φ)

 

Properties of unit impulse (d[n])

1. d[n] plays similar role to d(t) in continuous time

2. any sequence can be written as

   x[n]=k에대한sigma(d[k]x[n-k])=sigma(x[n]d[n-k])

3. each impulse’s value corresponds to the signal value of the original discrete-time sequence (x[n])

4. however unlike continuous-time case, we don’t need to deal with infinities or singularity caused by d[t] which made analysis difficult

5. a sequence can be viewed as scaled, shifted and superposed version of unit impulse

6. this representation is a basis for discussion on convolution, filtering and lots of transforms in discrete LTI systems.

 

exponential signals : x[n]=a^n

it is eigenfunctions to discrete LTI systems.

   (a^n -> k*a^n)

if a=e^jw, complex exponential (x[n]=e^jwn) we would have discrete time complex sinusoids.

a complex sinusoid of frequency w : x[n] = Ae^(j(wn+δ)) = |A|cos(wn+δ)+j|A|sin(wn+δ)

 

sinusoidals in discrete-time (사인 모양으로 움직이는 그래프)

periodic : x[n] = x[n+N] every n for some integer N

difference between continuous/discrete sinusoidal

   1. cos[wn] may not be a periodic signal depending on the value of w.

   2. different values of w my represent the same signal, in other words, cos[wn] and cos[w’n] may be exactly same signal for w’!=w (w=w’+2πr)

1. in order for cos[w(n+N)] = cos[wn], 모든 n에 대해서 wN=2πk for some integer k. (w는 π가 곱해진 유리수여야 period N)

2. discrete sinusoid cos[wn] for some w’ = w+2πr for any integer r will give cos[w’n]=cos[wn] (w가 0과 2π 또는 -π와 π 사이이면 어떤 sinusoid든지 표현이 가능하다.)

 

discrete-time systems

discrete input -> system -> discrete output

system operator : T{}

y[n] = T{x[n]}

 

Attribute of system

memoryless : output depends only on the current value of inputs

linear : scaling(*) + superposition(+)

   T{ax1[n]+bx2[n]}=ay1[n]+b2[n]

time-invariant : T{x[n-k]} = y[n-k]

   shift to input only causes the same shift to the output

causality : 현재와 과거의 input으로 output이 결정됨.

stability : BIBO(bounded input, bounded output).

   finite ouput for finite input

 

impulse response

output of a system T with respect to an input of unit impulse as the impulse response of the system.

   h[n] = T{δ[n]}

impulse reponse can completely specify LTI system.

 

discrete time LTI systems : linear and time-invariant

y[n] = ~무한 k 부터 무한까지 sigma (x[k]h[n-k])

 

Example of some LTI system

Delay system : y[n] = x[n-nd]

   h[n] = δ[n-nd]

Accumulator : y[n] = k에 대해 –무한대~n sigma (x[k])

   h[n] = u[n]

Difference : y[n] = x[n] - x[n-1]

   h[n] = δ[n] - δ[n-1]