Recording - Sampling

Question 1

What is the A/D Conversion?

Answer 1

= Analog to Digital Conversion
gather sensed data f’
change it to REPRESENTATION (way) f suited for digital processing

Question 2

What are the steps of A/D conversion?
+ short explanation
+ Approache

Answer 2

1. Sampling
   - (equidistant) steps/points where (continuous) amplitude/function values are measured
   - Fourier Series
2. Quantization
   - (not equidistant) finite number of numbers the values are represented
   - Probability theory

Question 3

What is the general form of the Fourier Series?

And what are its attributes?

Answer 3

Decomposition of a periodic function f ( t ) with period T > 0 into sine and
cosine functions
Used to represent a periodic function by a discrete sum of complex exponentials
Periodic function 􏰕-> converts into a discrete exponential or sine and cosine function
Non-periodic function 􏰕-> not applicable

• If f ∈ C1 (continuous and piecewise continuously differentiable):
  the Fourier series converges pointwise and uniformly.
• If f ∈L2([c, c+T])(square-integrable):
  the Fourier series converges: lim ∥fn (t ) − f (t )∥ = 0

Question 4

Notes to the complex Fourier Series?

Answer 4

• frequently: c = −T2
• only one (complex) coefficient ck
• complex amplitude contains phase information
• integral of the exponential function easy to compute

Question 5

What is the Gibbs phenomenon?

What is it? What is it doing?

Answer 5

What is it? Fourier series of a piecewise continuously differentiable periodic function / Fourier-Reihe einer stückweise kontinuierlich differenzierbaren periodischen Funktion
What is it doing? peculiar manner at jump discontinuities / eigentümliches Verhalten bei Sprungdiskontinuitäten
large oscillations of the n-th partial sum of the Fourier series / große Abweichungen der n-ten Partialsumme der Fourier-Reihe
maximum of the partial sum even above that of the function itself / Maximum der Partialsumme sogar über dem der Funktion selbst

Question 6

What is the Fourier Transformation?

What is it doing?

Answer 6

Used to represent a general, non-periodic function by a continuous superposition (integral) of complex exponentials
Periodic function 􏰕-> converts its Fourier series in the frequency domain
Non-periodic function 􏰕-> converts it into continuous frequency domain

Spectral characterization of non-periodic functions, FourierseriescoefficientsofaperiodicfunctionaresampledvaluesoftheFouriertransformofone period of the function / Spektrale Charakterisierung von nichtperiodischen Funktionen, Fourier-Reihenkoeffizienten einer periodischen Funktion sind abgetastete Werte der Fourier-Transformation einer Periode der Funktion
Converts time function into frequency domain function

Question 7

What is the behaviour of the time and frequency domain when comparing?

Answer 7

The narrower a function is in the time domain, the wider it is in the frequency domain (and vice versa)
There is no function (in L2) that is both limited in the time/spatial domain and the frequency domain

Question 8

What ist the Discrete-Time Fourier Transform (DTFT)?

Answer 8

It produces a periodic function of a frequency variable
Es ist eine lineare Transformation aus dem Bereich der Fourier-Analysis. Sie bildet ein unendliches, zeitdiskretes Signal auf ein kontinuierliches, periodisches Frequenzspektrum ab, welches auch als Bildbereich bezeichnet wird.

Question 9

What is the Discrete Fourier Transform (DFT)?

Answer 9

measurements of a period of a periodic signal, transformed into an M-periodic sequence of complex numbers
Sie bildet ein zeitdiskretes endliches Signal, das periodisch fortgesetzt wird, auf ein diskretes, periodisches Frequenzspektrum ab, das auch als Bildbereich bezeichnet wird.

Question 10

What is the interpretation of the DFT?

Answer 10

1. Discrete analogy of the formula for the coefficients of a Fourier series
2. Cross-correlation of the input sequence, fj, and a complex sinusoid at frequency μ/M (matched filter)
3. Complete description of the discrete-time Fourier transform (DTFT) of an M-periodic sequence, which comprises only discrete frequency components
4. Uniformly spaced samples of the continuous DTFT of a finite length sequence

Question 11

What is the Nyquist-Shannon-Theorem?

Answer 11

It is used to reconstruct the original signal by using an interpolation

Question 12

What is aliasing?

Answer 12

Undersampling - losing information of the original signal

Aliasing occurs if . . .
• the signal contains frequencies higher than half the sampling frequency
• high frequency noise is present
Solution
• low-pass filter the signal prior to digitization