Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 1 DISCRETE SIGNALS AND SYSTEMS Sequences Exponential Sequences s is a complex number If s is a real number and a=e then the sequence is called real exponential

Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 2 DISCRETE SIGNALS AND SYSTEMS Sequences Geometric Sequence: real exponential sequence defined as:

Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 3 DISCRETE SIGNALS AND SYSTEMS Sequences Sinusoidal sequence

Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 4 DISCRETE SIGNALS AND SYSTEMS Properties of Sequences Sum of two signals: w = x + y w(k) = x(k) + y(k) Multiplication of two signals: w = x. y w(k) = x(k) y(k) Multiplication of signal by a scalar: w = c x w(k) = c x(k) Energy of signal: If a signal is delayed by m time units then x(k) becomes x(k-m) Sequence: sum of scaled, delayed unit samples

Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 5 DISCRETE SIGNALS AND SYSTEMS Properties of Sequences Example:

Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 6 DISCRETE SIGNALS AND SYSTEMS Signal Measures The signal norm is defined: Some properties of the signal norm are: Norm 1: Sum of the magnitudes of each signal sample It is used to determine system stability

Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 7 DISCRETE SIGNALS AND SYSTEMS Signal Measures Norm 1: Sum of the magnitudes of each signal sample. It is used to determine system stability. Norm 2: Provides a measure of the signal power. It is the most frequent used measure. Norm infinity: Gives the peak magnitude of the signal.

Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 8 DISCRETE SIGNALS AND SYSTEMS Linear, Shift-Invariant Systems Discrete–time system: Converting input sequence x=x(n) into output sequence y=y(n) through transformation φ[.] y(n) = φ[x(n)] Α linear system is defined by the principle of superposition. If, Then a system is linear if and only if,

Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 9 DISCRETE SIGNALS AND SYSTEMS Linear, Shift-Invariant Systems Example 1: Is the following system linear? y(n) = 10x(n) - 5y(n-1) ___________________________________________ ____________________________________________ Yes, the system is linear

Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 10 DISCRETE SIGNALS AND SYSTEMS Linear, Shift-Invariant Systems Example 2: Is the following system linear? _________________ ______________________________________ No, the system is not linear

Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 11 DISCRETE SIGNALS AND SYSTEMS Linear, Shift-Invariant Systems A system is time–invariant or shift- invariant if, y(n) is response to x(n) then y(n-k) is response to x(n-k) : a signal delay of k samples Example 1: Is the following system shift-invariant? y(n) = 10x(n) - 5y(n-1) ________________________ _________________________ Yes, the system is shift-invariant

Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 12 DISCRETE SIGNALS AND SYSTEMS Linear, Shift-Invariant Systems Example 2: Is the following system shift-invariant? y(n) = n x(n) ______________ __________ No, the system is not shift-invariant We said that we can express: The system output response is:

Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 13 DISCRETE SIGNALS AND SYSTEMS Linear, Shift-Invariant Systems If the system is linear, the response of the system to a sum of inputs is the same as the sum of the system’s responses to each of the individual inputs: By definition: φ[δ(k)] = h(k) If the system is shift-invariant: φ[δ(n-k)] = h(n-k)  If a system is linear and shift-invariant, the convolution sum applies y(n) = x(n) * h(n)

Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 14 DISCRETE SIGNALS AND SYSTEMS Linear Convolution The graph method of computing the Convolution sum  Folding one of the sequences x(n) or h(n) over the horizontal axis and getting x(-k) or h(-k)  Shifting the folded sequence creating x(n-k) or h(n-k)  The addition of the product of the two sequences at time n yields the output y(n) Example: What is the response y(n) if h(n)={1,2,3} and x(n) = {3,1,2,1}

Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 15 DISCRETE SIGNALS AND SYSTEMS Linear Convolution As a result, y(n)={3,7,13,8,8,3}  If x(n): N samples, h(n): M samples, y(n): N+M-1 samples

Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 16 DISCRETE SIGNALS AND SYSTEMS Stability and Causality A system is stable if a bounded input produces a bounded output. Necessary and sufficient condition This is the norm as defined in a previous session Example: is the following system stable: y(n) = x(n) + b y (n-1) Calculating the norm we have: The system is stable

Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 17 DISCRETE SIGNALS AND SYSTEMS Stability and Causality A causal system is a system that at time m produces a system output that is depended only on current and past inputs, that is: n<m This is always true for a unit impulse response it is zero for n<0  A discrete-time, linear, shift-invariant system is causal if and only if h(n)=0 for n<0 Example: is the following system causal: y(n) = x(n) + b y (n-1) Since the unit-sample response is zero for n<0, the system is causal.

Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 18 DISCRETE SIGNALS AND SYSTEMS Digital Filters A broad class of digital filters are discribed by linear, constant coefficient, difference equations {a i } {b i } characterize the system Given: initial conditions x(i), y(i) i=-1,-2,…,-M input sequence: x(n) output sequence: y(n) The system is causal. The system is Mth-order. Two main classes of digital filters:  Infinite Impulse Response (IIR)  Finite Impulse Response (FIR)

Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 19 DISCRETE SIGNALS AND SYSTEMS Digital Filters  Infinite Impulse Response (IIR): current and past input samples and past output samples. Example: Determine impulse response for the first- order IIR filter. y(n) = x(n) + b y (n-1) Assume: x(n)=0, y(n)=0 for n<0 x(n)=δ(n) h(n)=δ(n) + b h(n-1) h(n)=0 n<0 h(0)=1 + b 0 = 1 h(1)=0 + b 1 = b h(2)=0 + b b = b 2 h(3)=0 + b b 2 = b 3 …….. h(n)= b n u(n)

Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 20 DISCRETE SIGNALS AND SYSTEMS Digital Filters  Finite Impulse Response (FIR): current and past input samples.  The coefficients of the FIR filter are equivalent to the filter’s impulse response. Why? Remember convolution? h(k) = b k k=0,1,2,…,M h(k) = 0 otherwise