Echivalarea sistemelor analogice cu sisteme digitale

An analogic system is composed from circuits A digital system is a computer algorithm

(continuous time) signal An analogic (continuous time) signal Digital signal, a numerical sequence, obtained by sampling and AD conversion

Constant signal with digital frequency 0 cycles/sample The maximum digital frequency is 0.5 cycles/sample

There are some “problems” with sampling There are some “problems” with sampling. The signal having a 0 digital frequency can be obtained from many analogic signals The signal with 0.5 digital frequency can be also obtained from many analogic signals

Sampling Sampling – time-domain discretization The Sampling Theorem Ideal Sampling.

A sample of x(t) is obtained multiplying the continuous-time signal x(t) with the rectangular impulse uΔ(t ): Another sample can be obtained using the same rectangular impulse shifted by kTs.

sampling process : small values of Δ: ideal sampling of the signal x(t) :

Ideal Sampling Mathematical model System model

Spectrum of Ideal Sampled Signal the spectrum of the ideal sampled signal :

The Fourier transform of a product is convolution of the Fourier transforms. The Fourier transform of the periodic Dirac’s distribution is also a periodic Dirac’s distribution. The effect of the convolution of a specified function with the periodic Dirac’s distribution is the periodic repetition of the considered function.

The spectrum of an ideal sampled signal is the periodic repetition of the spectrum of the original signal. The period is inverse proportional with the sampling step Ts.

Spectrum of original signal Spectrum of periodic Dirac distribution Spectrum of ideal sampled signal Aliasing error

Sampling Band-limited Signals x(t)-band-limited

successive replica of the original signal spectrum are not superimposed and the spectrum of the original signal can be recovered by low-pass ideal filtering for

Ideal Low-pass Filter

the aliasing error can be avoided. For perfect reconstruction : Sampling freq. Cutoff freq for the low pass filter

The frequency response of the reconstruction filter is: Its response : with the spectrum:

When the condition: is not verified, the aliasing error appears.

Reconstruction

WKS (Whittaker, Kotelnikov, Shannon) Sampling Theorem If the finite energy signal x(t) is band limited at ωM , ( X(ω)=0 for |ω | > ωM), it is uniquely determined by its samples if the sampling frequency is higher or equal than twice the maximum frequency of the signal: the original signal can be reconstructed from its samples a.e.w: if the cut-off frequency of ideal low-pass reconstruction filter :

The link between the two frequency axes of spectra of the discrete signal and of the corresponding continuous signal is:  spectrum periodic The maximum frequencies are also related:

Ideal Low-pass Filtering Reconstruction The signal reconstructed from curves of type sin x / x. Interpolation

Linear interpolation approximates the signal using straight lines that unify points determined by the samples Reconstruction filter is triangular. errors

Reconstruction by Zero Order Extrapolation

The frequency response of the reconstruction filter is not perfectly flat in the pass band.

The Fourier Transform of Distributions 1) The spectrum of the Dirac’s distribution for any test function (t): the Dirac’s distribution is even. Hence, we have obtained:

2) The spectrum of the constant 1(t) duality

Approximation of continuous-time systems with discrete-time systems The continuous-time systems are replaced by discrete-time systems even for the processing of continuous-time signals. Impulse invariance method Step invariance method Finite Difference Approximation (FDA) Bilinear Transform

General Method for Quality Evaluation of Approximation for Band-limited Systems AD converter – Analog to Digital Converter DA converter –Digital to Analog Converter

Identification of the impulse response of the system h(t)

Identification of the impulse response of a band-limited system using the cardinal sine. Frequency response of the band-limited system that must be identified Spectrum of the input signal (cardinal sine) Spectrum of the output signal

1. Impulse Invariance Method

Change the input signal : changes the digital system. No perfect approximation. Approximation using “standard” signals: Unit step (t) Ramp signal t (t).

Approximation by sum of unit step signals, shifted and weighted

Approximation by sum of ramp signals, shifted and weighted

Impulse invariance method for digital systems equivalent with band-limited c.t. systems

Relation between s and z planes for the Impulse Invariance Method Left half plane <0  interior of unit disc |z|=r<1 Imaginary  unit disc |z|=1 axis =0 Right half plane >0  exterior of unit disc |z|=r>1 segment [-/T, /T) on imaginary axis  one wrapping on the unit disc

Avoid aliasing errors from the frequency response of the digital system obtained: frequency response of the analog system: completely included in freq. band -π/T , π/T. band-limited analog system Sampling freq.

Impulse invariance method: link between frequency responses of equivalent systems Transfer functions Freq. responses The frequency response of the digital system is the same with the frequency response of the analog system of limited band for frequency less than half of sampling frequency

frequency response of the digital system ~ comb

Example: RC circuit non band limited system  aliasing errors

Unit step response

Frequency response

2. Step Invariance Method Input signal: unit step Step response of the analog signal Digital input signal:

Digital system:

Example: RC circuit

3. Finite difference approximation First derivative approximation Transfer function of the digital system:

General case

Relation Between s and z planes Finite Difference Approximation

s-plane | z-plane

Stable analog system => stable digital system Analog system and digital system: identical freq. response if imaginary axis on s-plane = unit circle on the z-plane. Not true for this method!!!

Example: first order LPF

Finite difference approximation

Finite difference approximation

Finite difference approximation higher accuracy !

4. Bilinear Transform Input signal Area An (ABCD)~ integral In: numerical methods (trapezoidal rule)

Bilinear transform:

Relation of the s and z planes for the bilinear transformation method

Relation between frequency responses for the bilinear transform

Distorted digital freq Distorted digital freq.response due to non-linear relation between frequencies!!!