Linjära system: Seminarie 1

What is the difference between continuous-time and discrete-time signals?

Kontinuerlig tid: Signalen finns definierad för varje tidsögonblick, exempelvis x(t)

Diskret tid: Signalen finns endast vid diskreta tidpunkter, exempelvis x[n]

En diskret signal kan skapas genom sampling av en kontinuerlig signal

What is the difference between discrete-time and digital signals?

Diskret-tidssignal: Diskret i tid, men amplituden kan fortfarande ha godtyckligt många värden.

Digital signal: Diskret både i tid och amplitud.

Digitala signaler är alltså kvantiserade.

What does sampling of analog signals mean?

Sampling innebär att man mäter en analog signal med jämna tidsintervall:

x[n]=x(nTs)

där:

• Ts_s = samplingstid

• fs=1/T_s=1/T_s = samplingsfrekvens

Why use digital signal processing of analog signals?

Digital signalbehandling används eftersom den:

• är programmerbar och flexibel

• kan implementeras exakt

• är lätt att lagra och överföra

• ofta är billigare och stabilare

• möjliggör avancerade filter och algoritmer

What is a sequence?

En sekvens är en följd av värden indexerade med ett heltal:

x[n],n∈Z

Exempel:

x[n]={1,2,3,4}

What is the difference between finite-duration and infinite-duration sequences?

Finite-duration vs infinite-duration sequence

• Ändlig sekvens: Har ett begränsat antal icke-nollvärden.

• Oändlig sekvens: Har icke-nollvärden över ett obegränsat antal index.

Exempel:

x[n]=u[n]−u[n−5]

är ändlig eftersom den endast är aktiv för 0≤n<5.

What is a periodic sequence?

En sekvens är periodisk om det finns ett positivt heltal N sådant att:

x[n+N]=x[n]

Det minsta sådana NN kallas grundperioden.

Are all sinusiodal sequences periodic?

Nej. För en diskret sinus:

x[n]=Acos⁡(ω0n+ϕ)

är den periodisk endast om:

ω0/2π

är ett rationellt tal.

Det måste alltså finnas heltal N,k,så att:

ω0​N=2πk

What is a bounded sequence?

What is an energy sequence and a power sequence?

What are the basic operations on sequences that we use?

What does right-shift and left-shift of sequences mean? How does it relate to time?

What is reversal of a sequence?

What is a discrete-time system?

What is a linear system?

What is a time-invariant system?

What is a causal system?

What is a stable system?

What is the impulse response of a system?

What is the step response of a system?

How to determine stability of a system, given the impulse response?

What is convolution?

What is the impulse response of a cascade connected system?

What is the impulse response of a parallel connected system?

What is a difference equation?

How can the difference equation be solved (that is, how can the output be computed?)

What are IIR and FIR systems?

What does synthesis of LTI systems mean?

What does realization/structure/algorithm of LTI systems mean?

Which building blocks are used in realizations of LTI systems?

What does implementation of LTI systems mean?

Why do we use/need Fourier series and Fourier transforms?

When can we use a Fourier series instead of the Fourier transform?

What is a main difference between the Fourier transforms of discrete-time and continuous-time signals?

What are amplitude spectrum and phase spectrum?

Why is it enough to study the Fourier transform in the frequency interval from 0 to pi for real-valued

sequences? (this is related to question 34)

What does the property linearity of the Fourier transform imply?

What happens in the Fourier transform domain when we shift a sequence?

What happens in the Fourier transform domain when convolv two sequences?

What is Parseval’s relation?

Why do we use/need the z-transform? (instead of only using the Fourier transform?)

What is the region of convergence? Why is it of interest?

What types of region of convergence do we have?

What are poles and zeros?

What can be said about the pole and zero locations for real-valued sequences?

How to compute inverse z-transforms without using integrals?

What does the property linearity of the z-transform imply?

What happens in the z-transform domain when we shift a sequence?

What happens in the z-transform domain when convolv two sequences?