DSP OHT 1 Cheat Sheet PDF

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Summary

This is formula sheet compilation by my batch mates Usama Mirza and Hamza Chippa for the course of Digital Signal Processing....

Description

∞

If ROC contains unit circle then the system is stable If ROC contains infinity then the system is causal ROC cannot contain any poles ROC must be a connected region

x[n] = ∑ x[k]δ[n − k] k=− ∞ ∞

u[n] = ∑ δ[n − k] k=0

δ [n] = u[n] − u[n − 1] N = 2πk ω I deal delay system : y [n] = x [n − nd ] 1 M1 +M2 +1

M oving Average System : y [n] =

Sampling ∞

s(t) = ∑ δ (t − nT s )

M2

n=− ∞

∑ x [n − k ]

∞

∞

xs (t) = xc (t)s(t) = ∑ xc(t)δ(t − nT s) = ∑ xc(nT s)δ(t − nT s)

k=− M 1

n=− ∞

n=− ∞

n

Accumulator System : y[n] = ∑ x[k]

S (jΩ) =

k=− ∞

2π Ts

C ompressor System : y [n] = x[Mn]

∞

∑ δ(Ω − kΩ s), Ω s = 2π /T

s

k=− ∞ ∞

X s (jΩ) =

1 X (jΩ) 2π c

Finite Impulse Response (FIR) Infinite Impulse Response (IIR)

1 * S(jΩ) = T s ∑ X c(j(Ω − kΩ s)) k=− ∞

Memoryless: Depends only on present value on n h[n] = 0, for n =/ 0 Causal: Doesn’t depend on future values of n h[n] = 0, for n < 0

Ωs ≥ 2Ω N , Ω N is the greatest frequency in x c(t)

Stable: Finite output for finite input ∞

∑ ∣h[n] ∣ has to be finite

n=− ∞

Aliasing if Nyquist’s criteria not satisfied

Linear System: A x1 [n] + B x2 [n] → Ay 1 [n] + By 2[n] Time Invariant: x[n − no ] → y [n − no ]

Convolution:

Original signal can be recovered using a low pass filter

∞

x[n] * h[n] = ∑ x [k]h[n − k ] k= −∞

x[n] * δ[n] = x[n], x[n] * δ[n − k] = x[n − k] Complex exponentials are eigenfunctions of LTI Systems H(ej⍵ ) is the eigenvalue of an LTI System ∞

Fourier Transform: X (ejω ) = ∑ x[n]e − jωn (Periodic with period = 2𝜋) n= −∞

Synthesis Equation: x[n] =

1 2π

∫ X(e j ω)e j ωndω 2π

Fourier Series

Fourier exists if x[n] is absolutely summable

ak =

∞

Z-Transform: X (z) = ∑ x[n]z −n n= −∞

Synthesis Equation: x[n] =

Finite signal

Right sided signal

Left sided signal

Double sided signal

1 N

∑ x[n]e −jkω on Periodic with Period N n=

x[n] = ∑ ak ejkωo n 1 2πj

k=

∮ X (z )z n −1 dz C

Entire z plane except possibly z=0 or z=∞

Table 5: Properties of the Discrete-Time Fourier Transform 1 Z X(ejω ) ej ωndω x [n] = 2π 2π +∞ X x [n]e−jωn X(ejω ) =

Table 2: Properties of the Discrete-Time Fourier Series X X ak ejk (2π/N )n x [n ] = ak ejkω0 n = k=

1 X

ak =

Property

Time Scaling

x [n]e−jkω0 n =

n=

1 X

N

x [n]e−jk (2π/N )n



Fourier series coefficients

ak bk

Periodic with period N and fundamental frequency ω 0 = 2π/N

Ax [n] + By [n] x [n − n0 ] ejM (2π/N)n x [n] x ∗ [n ] x [−n]  x [n/m] if n is a multiple of m x (m) [n] = 0 if n is not a multiple of m



Periodic with period N

Aak + Bbk ak e−jk (2π/N )n 0 ak−M ∗ a−k a−k viewed as 1 ak periodic with m period mN

X

Aperiodic Signal

!

Convolution Multiplication

x [n]y[n]

Differencing in Time

x [r]y[n − r]

Nak bk X

x [n]y[n]

Fourier transform

x [n] y[n] ax [n] + by[n] x [n − n0 ] ej ω0n x [n] x ∗ [n] x [−n]  x [n/k ], if n = multiple of k x (k) [n] = 0, if n 6= multiple of k x [n] ∗ y[n]

Time Expansions

r=hNi

Multiplication

Property

Linearity Time-Shifting Frequency-Shifting Conjugation Time Reversal

(periodic with period mN )

Periodic Convolution

x [n] − x[n − 1] n X x[k]

Accumulation

Running Sum

x [n] − x [n − 1] n  X finite-valued and x[k] periodic only if a = 0 0 k=−∞

Conjugate Symmetry for Real Signals

x [n] real

Real and Even Signals

x [n] real and even

Real and Odd Signals

x [n] real and odd

Even-Odd Decomposition of Real Signals

(1 − e−jk (2π/N ) )ak   1 ak (1 − e−jk (2π/N ) )  ∗ a = a k −k   ℜe{ak } = ℜe{a−k } ℑm{ ak } = −ℑm{ a−k }  | ak | = | a−k |   0) or ∞ (if m < 0)

Differentiation in the z-Domain

Pn

1 1−z −1 X(z)

nx[n]

z) −z dX( dz

k=−∞ x[k]

Initial Value Theorem If x[n] = 0 for n < 0, then x[0] = limz →∞ X(z)

π

+∞ X

π j

+∞ X

Fourier series coefficients (if periodic) ak 2πm = N 1, k = m, m ± N, m ± 2N , . . . 0, otherwise irrational ⇒ The signal is aperiodic ω0 = 2πm N  1 k = ±m, ±m ± N, ±m ± 2N, . . . 2, ak = 0, otherwise ω0 irrational ⇒ The signal is aperiodic 2π ω0 = 2πr  N1 k = r, r ± N, r ± 2N , . . .  2j , 1 − 2j = , k = −r, −r ± N, −r ± 2N, . . .  0, otherwise ω0 irrational ⇒ The signal is aperiodic 2π 1, k = 0, ±N, ±2N , . . . 0, otherwise

(a)

ω0 ak

{δ(ω − ω0 − 2πl) + δ(ω + ω0 − 2πl)} (b) (a) {δ(ω − ω0 − 2πl) − δ(ω + ω0 − 2πl)}

ak

l=−∞

(b) 2π

+∞ X

δ(ω − 2πl)

Periodic  square wave 1, |n| ≤ N1 0, N1 < |n| ≤ N/2 and x[n + N ] = x[n] +∞ X δ[n − kN ] x[n] =

2π

+∞ X

  2πk ak δ ω − N

k=−∞

2π N

k=−∞

ak =

k=−∞



δ ω−

2πk N



1 1 − ae−jω sin[ω (N1 + 1 )] 2 sin(ω/2)  1, 0 ≤ |ω| ≤ W X(ω) = 0, W < |ω| ≤ π X(ω)periodic with period 2π

an u[n], |a| < 1 

1, |n| ≤ N1 0, |n| > N1   sinc Wπn 0 0

9. [cos ω0n]u[n]

1−[cos ω0 ]z−1 1−[2 cos ω0 ]z−1 +z −2

|z| > 1

At least the intersection of R and |z| > 1

10. [sin ω0n]u[n]

[sin ω0 ]z−1 −2 1−[2 cos ω0 ]z−1 +z

|z| > 1

11. [r n cos ω0n]u[n]

1−[r cos ω0 ]z−1 −2 1−[2r cos ω0 ]z −1 +r 2 z

|z| > r

12. [r n sin ω0n]u[n]

[ r sin ω0 ] z 1−[2r cos ω0 ]z −1 +r 2 z−2

|z| > r

R

−1

Useful Infinite Summation Identities (|a| < 1) ∞ X k=0 ∞

X

ak =

1 1−a

kak =

a 2

(1 − a) X 2 k a2 + a k a = (1 − a)3 k=0 k=0 ∞...