2019年最新-Fourierseries,transform,andFFT傅立叶级数,变换,FFT-精选文档

1、Analog and Digital Filter DesignFourier series, transform, and FFTMiroslav Lutovac and Dejan losicOverview Fourier series Fourier transform FFT Examples MATLAB fft MATLAB freqzFourier seriesPeriodic signals Nonsinusoidal periodic signals play an important part in signal processing Theory of nonsinus

2、oidal periodic signals is based upon resolving them into sinusoidal components According to the principle of superposition, we can find the steady-state response of an LTI system to an arbitrary periodic in put by applying the phasor method to harmonic comp on entsSecti on ally continu ous signal Co

3、nsider a real signal x(r) of real variable t that satisfies the following conditions: x(t+T) = x(r); that is, the signal is periodic having a period T x(t) is defined in the interval t<?<t + T x(r) and dx(r)/dr are sectionally continuous iriT<r<T + TDefinition of Fourier series+OGx(r) =

4、C()+(A/? cos(3/j + Bn sin(nir)/?=it + Tx(t) cos(ei/j dr2 Ct+t2ttThe process of resolving a signal into its Fourier series is called spectral analysis or harmonic analysisBn = J x(r)sin(Hir)drComplex Form of Fourier Series+ 00/?=-oo1 f r+r Cn = fT*IInParseval's Identityi pi+Ti +°°+°

5、;° /= & + 辽 + 晞=£ IQ|2&/Z= 1H = OC'lim Afl = 0moclim Bn = 0mooIf the signal x(t) represents an electric current or voltage across a resistor,the expression is proportional to the resistor average power over a periodHarmonics of periodic signals+OGx（t）=工兀叫）=X（°）/?=（）+0G+ T

6、X絆 cos(gf + $(,/)n=ide componentx(o)(r) = X(°)= C()Gibbs phenomenonWhen a sudden change of amplitude occurs in a signal and the attempt is made to represent it by a finite number of terms in a Fourier series, the overshoot at the corners (at the points of abrupt change) is always found. As the

7、number of terms is increased, the overshoot is still found; this is called the Gibbs phenomenon.x(/)Amplitude, phase, power spectrumy (2),A inA graphical representation of a periodic signal x(t) produced by drawing a series of vertical lines at intervals on a horizontal axis, where the intervals rep

8、resent an increase of namplitude spectrum花（°）, g,£，,凶），phase spectrumExample spectrumFourier tra nsformDefinitionFourier transformf +OGxg = J(x(r) = / x(r)e->fdrJ00Inverse Fourier transform1 C +°°M)=厂 1( X(x>)= / x c/WeZ 兀 J 0GThe variable co is called a continuous frequenc

9、y variablelim /|x(f)| df < OO sufficient existenee conditionTfgJ_TSpectrum of xPlots of |X(jco)| and arg(X(jco) versus co are called the amplitude spectrum and phase spectrum of x(t), respectivelyEn 言 ds 善tEe Egads as£dPropertiesUniquen essX(/j = X2(0 O Xi(/e) = X2(je)于(Kg) =Homogeneity于 t(K

10、 Xg) = K：L(X(j4)7M) + x2(t)= 7M) + JCv2(r)AdditivityJ_1( Xi()e) + X 2(丿 e) = J_1( Xi(je) + 厂1( 乂2(丿讪DifferentiationD心爭drJ(Dx(r) = ；J(x(r) = jcoX (jco)(j® X(j®) = D_1(= Dx (t)Parseval's theorem and energyspectral densityms =云Frequency response ofcontinuous-time systemsniny(0Dm = dei =0k

11、=0Relaxed LTI system described by linear constant-coefficients differential equationHO)Frequency responseDiscrete Fourier transform(FT)and FFTDefinitionx（）n=兀（）,兀（1）,兀（2）, ,兀（刃）, ,兀（N1）Finite-lengthsequenceN1Xg =工 厂"/?=()X伙）“ = X（）, X（i）, 乂,X（灯,X（“_i）In verse discrete Fourier transform(idft)N_1

12、%)=亓工闷斤=()Discrete Fouriertransform pairx()w 宀乂伙)®Q%)n = X(r)nQ1X)n = U(/?)/v< o 6>©66 6ro£0to9090zo(XIW!=9 乙039101eeeeeooParsevas identityN 1 N-lE l)|2 =万刀险|2/?=() £=()Frequency response ofdiscrete-time systemsMLQURelaxed LTI system described by linear constant-coefficient

13、s difference equationThe system transforms the input sequence by multiplying each member of the input DFT with the factor Hk)/?=（）/=（）Digital frequencyeInNormalized frequency(MATLAB)Frequency responseMATLAB freqz0.1freqz(叫町 b2 l>3, a。0 屯】)0.20.30.40.50.60.70.80.91Normalized Frequency (xn rad/sample)A o o o o o o o o 1 2 3 4 5 6 ffip) apn-E6es0.10.20.30.40.50.60.70.80.91Normalized Frequency (xn rad/sample)o2L(saaB8p) aseqdFurther readi ngM. D. Lutovac, D. V. Tosic, B. L. Eva nsHLTER DESIGN