Digital Signal Processing Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Intelligent Systems Laboratory Digital Signal Processing Prof. George Papadourakis, Ph.D.

Discrete Signals and Systems Sequences Discrete time signals are number sequences x is a sequence x(n) is a sample of the sequence at time n

Discrete Signals and Systems Sequences Creation of a sequence Number creation and place them in a sequence 1,2,3,…,{N-1} x(n) = n 1<n<N Iterative process x(n) = x(n-1)/2, x(0) = 1 3. Sampling of analog signal. Signal value create a sequence A/D converter 1,2 time independent, 3 time dependent

Discrete Signals and Systems Sequences Discrete function delta δ(n) It has one non-zero element Very important input sequence in a digital system. The output sequence is called impulse response and contains important information concearning the behavior of the system. Discrete function delta delayed δ(n-k)

Discrete Signals and Systems Sequences Unit step u(n) u(n) is related to δ(n) u(n) – u(n-1) = δ(n)

Discrete Signals and Systems Sequences Exponential Sequences : x(n) = Aasn = Aa(σ + jw0)n s is a complex number s = (σ + jw0) If s is a real number and a=e then the sequence is called real exponential x(n) = Ae-|σ|n

Discrete Signals and Systems Sequences Geometric Sequence : real exponential sequence defined as :

Introduction to Neural Networks Discrete Signals and Systems Sequences Sinusoidal Sequence :

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 = xy w(k) = x(k)y(k) Multiplication of signal by a scalar : w = cx w(k) = cx(k) Energy of the signal : If a signal is delayed by m time units then x(k) becomes x(k-m) Sequence : Sum of scaled, delayed unit samples :

Discrete Signals and Systems Properties of Sequences Example :

Discrete Signals and Systems Signal Measures The signal norm is defined : Some properties of the signal norm are:

Discrete Signals and Systems Linear Norm 1 : Sum of the magnitudes of each signal sample 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.

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[.] A linear system is defined by the principle of superposition. If, Then a system is linear if and only if,

Discrete Signals and Systems Linear Shift-Invariant Systems Example 1 : Is the following system linear? y(n) = 10x(n) – 5y(n-1) Solution : αφ[x1(n)] = α10x1(n) – α5y1(n-1) bφ[x2(n)] = b10x2(n) – b5y2(n-1) αφ[x1(n)] + bφ[x2(n)] = α10x1(n) – α5y1(n-1) + b10x2(n) – b5y2(n-1) φ[αx1(n) + b x2(n)] = 10[αx1(n) + bx2(n)] – 5[ αy1(n-1) + by2(n-1)] Yes, the system is linear!

Discrete Signals and Systems Linear Shift-Invariant Systems Example 2 : Is the following system linear? y(n) = [x(n)]2 Solution : αφ[x1(n)] = α[x1(n) ] 2 bφ[x2(n)] = b[x2(n) ] 2 αφ[x1(n)] + bφ[x2(n)] = α[x1(n) ] 2 + b[x2(n) ] 2 φ[αx1(n) + b x2(n)] = = [αx1(n) + bx2(n)] 2 = α2[x1(n)] 2 + b 2[x2(n) ] 2 +2αbx1(n)x2(n) No, the system is not linear!

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) ,z-k : a signal delay of k samples Example 1 : Is the following system shift-invariant? y(n) = 10x(n) - 5y(n-1) Solution: Yes, the system is shift-invariant!

Discrete Signals and Systems Linear Shift-Invariant Systems Example 2 : Is the following system shift-invariant? y(n) = nx(n) Solution : No, the system is not shift-invariant! We said that we can express : The system output response is :

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)

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}

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

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) + by(n-1) Calculating the norm we have : The system is stable!

Discrete Signals and Systems Stability and Causality A causal system is a system that at the time m produces a system output that is dependent 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) + by(n-1) Since the unit-sample response is zero for n<0.. The system is causal!

Discrete Signals and Systems Digital Filters A broad class of digital filters are described by linear, constant coefficient and difference equations {ai} {bi} 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)

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) + by(n-1) Assume : x(n) = 0, y(n) = 0 for n<0 x(n) = δ(n) h(n) = δ(n) + bh(n-1) h(n)=0 n<0 h(0)=1 + b 0 = 1 h(2)=0 + b b = b2 …….. h(1)=0 + b 1 = b h(3)=0 + b b2 = b3 h(n)= bn u(n)

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) = bk k = 0,1,2,…,M h(k) = 0 otherwise

Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Intelligent Systems Laboratory