Signals and SystemsEE 112Lecture 16: Discrete time Fourier series and transformKhosrow GhadiriElectrical Engineering Department Electrical Engineering Department San Jose State University© Khosrow GhadiriJean Baptiste Charles FourierJean Baptiste Charles Fourier (3/21/17685/16/1830)© Khosrow GhadiriJean Baptiste Charles Fourier (3/21/1768-5/16/1830)2Signal and System EE Dept. SJSUOutlinePrelude S-4Prelude S4 Discrete-time Fourier series representation S-6 Properties of Discrete-time Fourier series S-9 Duality S-9 Even and odd S-11 Parseval’s theorem S-13 Representation of aperiodic signals S-14El S16Example S-16 Representation of aperiodic signals (2) S-22 Examples S-24Some common DTFT pairs S-29Some common DTFT pairs S29© Khosrow Ghadiri 3Signal and System EE Dept. SJSUPreludeA continues-time sinusoid and signals are periodic costsintA continuestime sinusoid and signals are periodic regardless of the value of . For the discrete-time sinusoid (or exponential ) are not periodic regardless of value .A i id i i di l if i ti l bcostsintcos njne2A sinusoid is periodic only if is rational number. Proof: Is periodic only if cos n2cos cosonN nIs periodic only if  Both and are integer, then2oNka rational numberkkoN is rational number When condition is satisfied. The period of the is given bya rational number2oN2 a rational numberN2Nk© Khosrow Ghadirithe is given by4Signal and System EE Dept. SJSUoNoNkPreludeTo compute we must choose the smallest value of that make kNTo compute , we must choose the smallest value of that make . an integer. Example: If , then the smallest value of that makejne4152 2 15 15Nk k k koN2oNkk is an integer 2, therefore42oNk k k 2151521522oNk k    A sinusoid is not a periodic, because0.92220 cos 0.9n22   A sinusoid is not a periodic, because is not a rational b22200.9 9kkk   cos 0.9n0.9 2n© Khosrow Ghadirinumber.5Signal and System EE Dept. SJSUThe discrete Fourier series and transformTopic Time Function Frequency FunctionFourier Series Continuous-PeriodicDiscrete-Non-PeriodicFourier Transform Continuous-Non-Periodic Continuous-Non-PeriodicZ Transform Discrete-Non-Periodic Continuous-PeriodicDiscrete Fourier TransformDiscrete-Periodic Discrete-Periodic© Khosrow Ghadiri 6Signal and System EE Dept. SJSUDiscrete time Fourier series representationA discrete time periodic signal with fundamental period is NfnA discrete time periodic signal with fundamental period is characterized byoNfnofnfnN The smallest value of for which this equation holds is the fundamental period. The fundamental frequency isAn periodic signal can be represented by discrete-time Fourier oN2ooNradsamplefnNAn periodic signal can be represented by discretetime Fourier series made up of sinusoids of fundamental frequency and its harmonic. As in the continuous time Fourier, we may use a trigonometric or an exponential form of the Fourier series fn2ooNoNexponential form of the Fourier series.  Because of its compactness and ease of mathematical manipulations the exponential form is preferable to the trigonometric. The exponential Fourier series consist of exponentials20,, ,oojjjneee3jj© Khosrow Ghadiriand so on. 7Signal and System EE Dept. SJSU3,... ,...oojjneeDiscrete time Fourier series representationDiscrete-time exponentials whose frequencies are separated by (or 2Discretetime exponentials whose frequencies are separated by (or integer multiple of ) are identical because That means that the harmonic is identical to the hi 222jknjnj kn jneeeeorN th2rthharmonic.  In other words, the first harmonic is identical to the harmonic, the second harmonic is identical to the harmonic, and so on. 1oNst2ndoN There are only independent harmonic and their frequency range over an interval (because the harmonics are separated by )  All the discrete-time signals are band-limited to a band from to .Because the harmonics are separated by there can be only 2ooN22NBecause the harmonics are separated by there can be only harmonics in this band. This band can be taken from 0 to or any other contiguous band of width . This means we may choose the independent harmonic 2ooN22oN© Khosrow Ghadiriover or over , or over or…8Signal and System EE Dept. SJSU01okN 12okN  1okNDiscrete time Fourier series representationA periodic discrete signal that is nonzero only for a finite number fnA periodic discrete signal that is nonzero only for a finite number of samples in the interval with fundamental period is given byoNfn210ojknNNkkfn ce01onN Where are the Fourier coefficients and are given by0kkc211jknNNkcfneN Setting in above equation0nN11Ncfn0k  Which indicate that equals the average value of over a period. The Fourier coefficient are often referred to as the spectral coefficients of fnkc0oncfnNocfn© Khosrow Ghadiricoefficients of 9Signal and System EE Dept. SJSUfnProperties Discrete Fourier seriesSince the discrete Fourier series is finite series in contrast to the Since the