Signals and Systems Basic SignalsLecture 2Khosrow GhadiriElectrical Engineering DepartmentElectrical Engineering DepartmentSan Jose State UniversityThe unit impulse functionThe unit impulse function AKA Dirac delta function is given as: The unit impulse function AKA Dirac delta function is given as: Very large, 00, otherwisettt1 The value at is very large, and for The duration is very short0t 0t0tt012312The duration is very short. The area is one. The unit impulse function is a mathematical model to represent signals 1 For any real number 0d  that are highly localized in time. ©Khosrow Ghadiri Signal and System 2The unit impulse functionVisualization of the unit impulse functionVisualization of the unit impulse function1ft2ft111t0t011221 For any real number 0d   ©Khosrow Ghadiri Signal and System 3The unit impulse functionThe unit impulse function: The unit impulse function: Very large, 00, otherwisettf=inline('(1/e)*((t>0)&(t<=e))','t','e');e1/100;e=1/100;t1=-1;t2=5;t=[t1,t2];fplot(f,t,1e-5,1000,'-',e);set (gca,'FontSize', 20)xlabel('t');xlabel(t);ylabel('p_\epsilon(t)');title('Unit impulse function,\epsilon=1/100')axis([t -0.1 1.1/e]) ©Khosrow Ghadiri Signal and System 4The Impulse FunctionThe impulse function AKA Dirac Delta function is formally defined in The impulse function AKA Dirac Delta function is formally defined in association with any arbitrary function as:   0ft tdt fft Where is any continuous function at .0t ft ©Khosrow Ghadiri Signal and System 5The unit impulse selectivity propertyThe unit impulse selectivity property AKA the sifting propertyThe unit impulse selectivity property AKA the sifting property, for continuous at 01,for discontinuous at 0ooooft ft tttftdtftftftt,2ooff f ©Khosrow Ghadiri Signal and System 6The unit impulse function samplig propertySampling property of the Impulse functionSampling property of the Impulse function oo ofttt ft ttft10ftt012312 ©Khosrow Ghadiri Signal and System 7The unit impulse function convolution propertyConvolution property:Convolution property:   *oo ottfttftdftt Convolution of two Impulse functionsIf is Impulse functionftIf is Impulse functionft11oottftt t t t The multiplication of two Impulse function is meaningless. ©Khosrow Ghadiri Signal and System 8The unit impulse scaling propertyScaling property of Impulse functionScaling property of Impulse function Where a is scalar constant. 1at ta Proof: For , substitute for .0a att0f at at d at a f at at dt f but 0fat tdt f0f at at d at a f at at dt f  then, ,so For , substitute for .aat t0a att0f at at d at a f at at dt f  1at ta ©Khosrow Ghadiri Signal and System 90f at at d at a f at at dt fThe unit impulse scaling propertybutbut then, , so0fat tdt faat t 1at ta and aat ta1ttatta ©Khosrow Ghadiri Signal and System 10The unit impulse functionUseful relations Useful relations 01toootttdtt  For is useful for representing derivatives of discontinuous functions. 0ot tfunctions. tdutdu tt tdt ©Khosrow Ghadiri Signal and System 11The unit sample sequenceThe unit sample sequence is discrete time version of the unit impulse The unit sample sequence is discrete time version of the unit impulse in continuous-time.Very large00otherwisenn Any arbitrary discrete function (sequence) can be written as summation of weighted unit sample sequences:0otherwise k is integer   kfnfknkg ©Khosrow Ghadiri Signal and System 12The Unit Step FunctionThe unit step function AKA Heaviside function is defined utThe unit step function AKA Heaviside function is defined mathematically by:1000tuttut The unit step function related to the unit impulse by:00ttut d The amplitude of unit step function is equal to 1 for all 0t utt ©Khosrow Ghadiri Signal and System 13t011The Unit Step FunctionThe unit step:The unit step:1000tuttf=inline('t>0','t');t1=-10;t2=10;t=[t1,t2];fplot(f,t);xlabel('t');ylabel('u(t)');>> title('Unit step function')>> axis([t -0.1 1.1]) ©Khosrow Ghadiri Signal and System 14The Unit Step FunctionThe product of any continuous time signal by units step xtThe product of any continuous time signal by units step function is equal to for and is equal to zero for xtxt0t  0t,0xtttt,0, 0xtutt ©Khosrow Ghadiri Signal and System 15The unit step functionThe unit-step function is equal to the integral of the unit utThe unit-step function is equal to the integral of the unit impulse ,forall 0tdt utt,for all 00, for 0tdtutdt   ©Khosrow Ghadiri Signal and System 16The unit step sequenceThe unit step sequence