Digital Signal Processing (DSP)

SIGNAL → mathematical model of phenomena

DIGITALIZING → translate a physical quantity into numbers.

SIGNAL → FUNCTION → NUMBERS

FILTER DESIGN → remove all we don't want from the signal

SYSTEM → mathematical formal model that transforms the signal in input (ex: filter)

ex: transform from the time domain into frequency domain (Fourier transform)

SCHEMA: blocchi in successione

SENSORS: transform a non-electrical quantity into an electr. quantity (ex: temperature)

The signal can be analog or digital (just converted by the sensor)

INPUT

SIGNAL CONDITIONING: adapt the signal → ex: improve

AND INTERFACING clean up the signal: amplification, ADC ...

DSP:

• digital control
• digital filtering
• transform
• parameter estimation (ex. estimation of position in a drone)
• feature extraction
• optimization

OUTPUT SIGNAL CONDITIONING & INTERFACING:

• D/A Conversion
• PWM modulation
• Amplification

DIGITAL ADVANTAGE:

• flexibility
• robustness
• predictability
• performance
• development time
• Known (numerical) precision

_{R1 & R2 are affected by tolerance}

_{R1 & R2 change with temperature & age}

^{gain G_o is G ± ΔG}

^{There's an offset in operational amplifier}

y = (1 +

^R₂

------------ x

_R1

SEQUENCES

The distance of the pulse depending on sampling time adimensional index

1. X(m) = δ(m) = { 1 m = 0 { 0 m ≠ 0

   Kronecker delta

   δ(t) is a distribution (∫_−∞^∞δ(t)dt = 1 di solito) δ_K(t) è diversa da δ(m)

2. Unit Step: X(m) = U(m) = { 1 m > 0 { 0 m < 0

3. Exponential Sequence: X(m) = Aα^mU(m) =

   • if 0 < α < 1 → va a zero
   • if -1 < α < 0 → si alterna e va a zero
   • if α > 1 → diverge
   • if α < -1 → diverge alternando

for n > M_o.

Al esempio se ci dobbiamo analizzare dati sobati prima si puo' usare un sistema causale un moto non causale

The moving average is causal system.

y_m = (1 / (2N+1)) ^N∑_K=-N x(m-K)

scritto con consider onde i campioni futuri

faccio la media rispetto al centro

ON-LINE (real time) PROCESSING

• processing the data as soon as they come.
• we want ready result.
• Computational speed must be high per restare nei tempi predefiniti → deadline
• es: ABS delle auto

OFF-LINE (batch) PROCESSING

• processing the data from a huge pack of data. The processing time is relativity important.

IH: an LTI system is BIBO stable

∑ |h(m)| < +∞

converge

PROOF:

|y(m)| = |∑_k=-∞^∞ x(m) h(m-K)| ≤ ∑_k=-∞^∞ |x(m)| |h(m-K)|

triangular inequality

∀ x(m) ∃ B |x(m)| ≤ B → |y(m)| ≤ B'

B ∑_M=-∞^∞ |h(m-K)| < +∞

stable system

PROPERTIES OF CONV. SUM

y(m) = ∑_k=-∞^∞ x(k) h(m-K)

in ideal form it can be used approssimally

1. if h(m) = 0 M < 0
2. if |x(m)| x(m) = 0 M < 0

y(m) = ∑_k=0^M x(K) h(m-K)

campa del tempo e spostata verso sinistra

CONSTANT-COEFFICIENT LINEAR DIFFERENCE EQUATIONS (CLDE)

x(m) → [T] → y(m)

∑_k=0^Na_ky(m-k) = ∑_k=0^Mb_kx(m-k)

• CONSTANT

linear combination of inputs and outputs values

→ y(m) = ∑_k=0^M b_k/a₀ x(m-k) - ∑_k=1^N a_k / a₀ y(m-k)

IF x(m)=0 M