THE CTFT OF SYSTEMS 1.1 2008/07/05 00:49:09.795 GMT-5 2008/07/05 01:07:04.093 GMT-5 Phuong Huu Nguyen nhphuong@hcmuns.edu.vn Phuong Huu Nguyen nhphuong@hcmuns.edu.vn Analog (continuous-time) systems are characterized by their impulse responses. In the time domain the output signal y(t) of the system is the convolution of the input signal x(t) with the system impulse response h(t): y ( t ) = x ( t ) ∗ h ( t ) size 12{y \( t \) =x \( t \) * h \( t \) } {} By the convolution theorem, the above equation is transformed into the Fourier domain as Y(F)=X(F)H(F) size 12{Y \( F \) =X \( F \) H \( F \) } {} where H(F), the Fourier transform of the impulse response h(t), is the frequency characterization of the system and is called frequency response. From above we write H(F)=Y(F)X(F) size 12{H \( F \) = { {Y \( F \) } over {X \( F \) } } } {} Now the frequency response can be interpreted as the ratio of the Fourier transform of the output signal to the transform of the input signal. The frequency response is, in general, complex and we write H(F)=∣H(F)∣ejΦ(F) size 12{H \( F \) = lline H \( F \) rline e rSup { size 8{jΦ \( F \) } } } {} where ∣H(F)∣ size 12{ lline H \( F \) rline } {} is the magnitude response and Φ(F) is the phase response. For example, for ideal analog systems the output y(t) with respect to the input x(t) is y(t)=Gx(t− t 0 ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadMhacaGGOaGaamiDaiaacMcacqGH9aqpcaWGhbGaamiEaiaacIcacaWG0bGaeyOeI0IaamiDamaaBaaaleaacaaIWaaabeaakiaacMcaaaa@4129@ where G is a scale factor (gain or attenuation), and t0 size 12{t rSub { size 8{0} } } {} is a time delay. The above equation is Fourier transformed to Y(F)=GX( F ) e −2πF t O MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadMfacaGGOaGaamOraiaacMcacqGH9aqpcaWGhbGaamiwamaabmaabaGaamOraaGaayjkaiaawMcaaiaadwgadaahaaWcbeqaaiabgkHiTiaaikdacqaHapaCcaWGgbGaamiDamaaBaaameaacaWGpbaabeaaaaaaaa@4529@ then H(F)=Y(F)X(F)=Ge−2πFt0 size 12{H \( F \) = { {Y \( F \) } over {X \( F \) } } = ital "Ge" rSup { size 8{ - 2π ital "Ft" rSub { size 6{0} } } } } {}

Ideal systems

Thus for ideal systems, the magnitude response |H(F)| is constant, independent of frequency, and the phase response Φ(F)=−2πF t 0 =−2π t 0 F MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabfA6agjaacIcacaWGgbGaaiykaiabg2da9iabgkHiTiaaikdacqaHapaCcaWGgbGaamiDamaaBaaaleaacaaIWaaabeaakiabg2da9iabgkHiTiaaikdacqaHapaCcaWG0bWaaSbaaSqaaiaaicdaaeqaaOGaamOraaaa@47C0@ is proportional to frequency F (), Such systems are called linear phase. For real systems ( i.e. systems having real-valued impulse responses), the magnitude response is symmetric, and the phase response is antisymmetric as for real signals ( Equation ).