VLAB Session 1

VLAB – Session 1

1

Digital Signal Processing

M. Ruggieri – T. Rossi


M. Ruggieri – T. Rossi 2

Generation of sequences

Unit impulse e unit step.Note: Although in theory we are required to define a signal in the [-∞ ,+∞] range, in Matlab we can represent only a part of a signal within a finite range as a vector with a finite number of elements.

Generation of the domain of our sequences, a vector whose values range from -10 to 20 (with a unitary step).

Generation of the values of the sequences, two vectors, one for the unit sample and one for the unit step.

‘stem’ is a function used to plot sequences (see Matlab help)



Complex exponential sequence:

Generation of sequences (cont’d)

Definition of the signal

Plot of the real and imaginary parts of the signal. ‘subplot’ is a function to create subplots inside a figure (see Matlab help)

Polar plot of the signal



Sinusoidal sequence:

Definition of the signal, the argument of ‘cos’ is in radians

What is “f”? “f” is a frequency but its unit is not Hz! It measures the cycles per sample!

Which is the period of this sequence?

What is theta?

Generation of sequences (cont’d)



Three-points moving average filter:

Random noise generation through ‘rand’ function (see Matlab help)

Is this a FIR or IIR filter?

What is the effect of this filter? This filter smoots the input signal.

A first LTI Filter Example

This filter can be used to reduce the noise of a signal.Lets define a signal and corrupt it with noise.

Signal generation

Signal corrupted by noise

Definition of the output of the filter

𝑦 𝑛 =1

3𝑥 𝑛 + 𝑥 𝑛 − 1 + 𝑥(𝑛 − 2)



Plot of filter input and oputput signals

A first LTI Filter Example (cont’d)

Consider a generic N-points moving average filter. What is the length of the filter?

What is the effect of an increase of filter lenght? --> The smoothness of the output increases! If the length of the filter is the same of the input the output is the input signal mean.

Calculation of the residual noise (after moving-average filtering)

Note: Matlab has built-in functions to define and manage LTI filters that have not been used here but will be introduced in the next examples



Consider the DT system defined by:

Definition of two input signals and their linear combination

Linearity property of a DT system

Lets check the linearity property:

The filter in matlab is defined through the coefficients of the linear constant coefficients equation

Outputs of filter for the three input signals

Filter output is calculated thorugh the ‘filter’ function (see Matlab help).



Linearity property of a DT system (cont’d)

Verification of the linearity property: if ‘d’ is equal to zero the system satisfies the property

Plot of the output signals and their difference

Note: the difference signal, ‘d’, is not exactly zero (but very close to it) due to calculation rounding and truncation effects.



Consider the DT system defined by:

Filter definition through the coefficients of the linear constant coefficients equation

LTI system impulse response

For an IIR filter it is not possible to comupte infinite samples of the inpulse response, hence Matlab function needs the number of inpulse response samples to be calculated as input.

Through the ‘impz’ function Matlab calculates the first N samples of the impulse response (see Matlab help)



Verification of filter stability property based on the sum of impulse response samples absolute values:

Recursive calculation of the sum of impulse response samples absolute values.The system impulse response samples modulus follows a decreasing trend, hence it can be supposed that if (for a certain value of ‘n’) a very low value, ε, is reached the summation converges. The value of εhas been set to 10-6

LTI system impulse response (cont’d)


M. Ruggieri – T. Rossi11

Calculation of linear convolution between a FIR filter impulse response and a finite-length sequence.

Matlab ‘conv’ function ca be used to coumpute linear convolution (see Matlab help).

Linear convolution computation

Sequences definition

Which is the length of the linear convolution?

Matlab ‘filter’ function ca be used to coumpute linear convolution. But we have to remember that the output length of ‘filter’ function is the same of the input signal, hence if we want the full values of the lienar convolution we have to use a zero-padding



The DTFT of a sequence is a continuous, complex and periodic function. It is not possible for a calculator to compute infinite values, hence we have to identify a sampling of the frequency response domain.

Filter frequency response

Matlab ‘freqz’ function can be used to calculate system frequency response (see Matlab help)

Is this a LP, HP, BS or BP filter?



Finite-length sequences definition

Convolution through DTFT

DTFT calculation for the two sequences

Convolution calculation in the discrete-time domain

Product of the two DTFTs

DTFT calculation for the DT domain convolution

Evaluate if the plot of ‘hp’ and ‘h3’ are identical



Time- and frequency-shifting properties of DTFT

Test sequence

Sequence DTFT

DTFT of time-shifted sequence

Check the differences between the two spectra

DTFT properties:

Plot of modulus and phase of the two DTFTs



Time- and frequency-shifting properties of DTFT

(cont’d)

Test sequence

Sequence DTFT

Modulated sequence and its DTFT

Check the differences between the two spectra

Plot of modulus and phase of the two DTFTs

Documents

VLAB Session 1