The continuous-time Fourier series (CTFS) This first section gives, a review of the continuous-time Fourier series (also called Fourier exapansion) of periodic signals . A periodic signal of fundamental frequency Ω 0 =2π F 0 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabfM6axnaaBaaaleaacaaIWaaabeaakiabg2da9iaaikdacqaHapaCcaWGgbWaaSbaaSqaaiaaicdaaeqaaaaa@3D91@ can be expanded into an infinite simusoidal and cosinusoidal series having frequencies which are the multiples of the fundamental frequency. There are three forms of expansion : Trigonometric ( Equation ) , amplitude and phase ( Equation ) , and complex exponential ( Equation ). Since the expansion coefficients (analysis equation) is complex ( Equation ), we have the magnitude spectrum | X n | MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaacYhacaWGybWaaSbaaSqaaiaad6gaaeqaaOGaaiiFaaaa@39E9@ and phase spectrum Φ n MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabfA6agnaaBaaaleaacaWGUbaabeaaaaa@387C@ or ∠ X n MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabgcIiqlaaysW7caWGybWaaSbaaSqaaiaad6gaaeqaaaaa@3B0A@ . Periodic signals have line spectrum (or discrete spectrum). In Fourier analysis we often come across the special function sin⁡x/x MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiGacohacaGGPbGaaiOBaiaadIhacaGGVaGaamiEaaaa@3B68@ (written for sin⁡(x)/x MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiGacohacaGGPbGaaiOBaiaacIcacaWG4bGaaiykaiaac+cacaWG4baaaa@3CC1@ ), also called sincx or Sa(x) ( Figure ). When we reconstruct the signal from its expanded harmonics we can get only an approximate waveform with ripples , especially at the abupt changes ( Figure ). The overshoot and undershout and ripples constitute the Gibbs effect (or Gibbs phenomenon).

Continuous-time Fourier transform (CTFT) Practical signals are mostly aperiodic whilst the Fourier series only applies to pesiodic signals . Figure shows the evolution from Fourier series of a periodic signal to the Fourier transform of the aperiodic one. As a result , we obtain the CTFT (analysis equation) ( Equation ) and the inverse CTFT (analysis equation) ( Equation ). The CTFT is complex , giving rise to a magnitude spectrum |X(F)| MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaacYhacaWGybGaaiikaiaadAeacaGGPaGaaiiFaaaa@3AE4@ and a phase spectrum Φ(F) MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabfA6agjaacIcacaWGgbGaaiykaaaa@3981@ . For real-valued signals the magnitude spectrum is symmetric and the phase spectrum is antisymmetric ( Equation ). The CTFT has many useful properties : Linearity, time shift, frequency shift (also called modulation theorem), time convolution ( also called convolution theorem), Parseval’s theorem. Time convolution means that the convolution of two time functions and the normal product of their Fourier transforms is a transform pair ( Equation ). There is also the convolution in frequency ( Equation ). In passing, we should note that the convolution of a signal with an unit impulse is just that signal ( Equation ). The Parseval’s theorem equates the signal energy in time domain to that in frequency domain. Section gives the CTFT of many basic signals : Narrow pulse δt MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabes7aKjaadshaaaa@3881@ , unit impulse δ(t) MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabes7aKjaacIcacaWG0bGaaiykaaaa@39DA@ , a constant , unit step, cosine and sine.

The CTFT of systems The CTFT also applies to continuous-time systems. The Fourier transform H(F) MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadIeacaGGOaGaamOraiaacMcaaaa@38D4@ of the system impulse response h(t) is called the frequency response of the system. By the time convolution it is shown that the frequency response is the ratio of the Fourier transform of the output signal and the transform of the input signal (Equation Equation ). Since the frequency response is complex ( Equation ), it has a maginitude response, and a phase response. Ideal analog systems have constant magnitude response (independent of frequency), and linear phase response (phase is porportional to frequency) (Equation ) and (Equation ).

Discrete-time Fourier series (DTFS) A sequence x(n) is periodic at period N indices (samples) according to Equation . Such a sequence can be expanded into a sersies of N frequency components (synthesis equation) as given by Equation . The spectral coefficients (analysis equation) is given by Equation . Notice the two basic differences between the DTFS and the continuous-time Fourier series.

Discrete-time Fourier transform (DTFT) We can evolve the DTFS having discrete spectrum to the DTFT having continuous spectrum. The transform (synthesis equation) is denoted X(ω) MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadIfacaGGOaGaeqyYdCNaaiykaaaa@39E6@ where ω is the digital angular frequency (radians/sample). The DTFT (analysis equation) is given by Equation . For nonperiodic sequence x(n), its DTFT is periodic in ω MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabeM8a3baa@37B0@ at period 2π MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaaikdacqaHapaCaaa@385C@ radians with the central period taken as [−π,π] MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaacUfacqGHsislcqaHapaCcaGGSaGaeqiWdaNaaiyxaaaa@3CBA@ or [0,2π] MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaacUfacaaIWaGaaiilaiaaikdacqaHapaCcaGGDbaaaa@3B86@ . Remember the frequency range [−π,π] MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaacUfacqGHsislcqaHapaCcaGGSaGaeqiWdaNaaiyxaaaa@3CBA@ corresponds to the Nyquist interval [− f s /2, f s /2] MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaacUfacqGHsislcaWGMbWaaSbaaSqaaiaadohaaeqaaOGaai4laiaaikdacaGGSaGaamOzamaaBaaaleaacaWGZbaabeaakiaac+cacaaIYaGaaiyxaaaa@4050@ with respect to the analog frequency F. The DTFT of a sequence exists if the sequence converges (absolutely summable). Usually the DTFT is a complex quantity (Equation ) having the magnitude spectrum |X(ω)| MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaacYhacaWGybGaaiikaiabeM8a3jaacMcacaGG8baaaa@3BE6@ and the phase spectrum Φ(ω) MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabfA6agjaacIcacqaHjpWDcaGGPaaaaa@3A83@ . For real-valued signal, |X(ω)| MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaacYhacaWGybGaaiikaiabeM8a3jaacMcacaGG8baaaa@3BE6@ is symmetric and Φ(ω) MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabfA6agjaacIcacqaHjpWDcaGGPaaaaa@3A83@ is antisymmetric (Equation ). Typical example is the DTFT of a digital symmetric rectangular pulse. The magnitude spectrum consists of a mainlobe and several sidelobes ( Figure ). Notice the imterpretation of the phase response. The formula of middle geometric series Equation and the trigonometric conversion Equation are useful.

Properties of DTFT The DTFT has many useful properties similar to the CTFT, namely: Linearity , time seversal , time shift, frequency shift , time convolution , frequency convolution , Parseval’s theorem. When a signal is shifted in time, its magnitude spectrum does not change but its phase spectrum becomes proportional to the frequency (Equation (3.53) and Figure(c) ).

DTFT of some popular signals First , as in many other transfroms , the DTFT of the unit sample δ(n) MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabes7aKjaacIcacaWGUbGaaiykaaaa@39D4@ is 1. Next, come the delayed unit sample, unit step, and symmetric rectangular pulse. Last is the complex exponential, cosine, and sine, the transforms of these signals are functions of frequency impulses.

Frequency response of LTI (LSI) systems As for continuous-time systems, the DTFT H(ω) MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadIeacaGGOaGaeqyYdCNaaiykaaaa@39D6@ of the impulse response is the frequency response . By the convolution theorem it can be shown that the frequency response is the ratio of the DTFT of the output signal and the DTFT of the input signal ( Equation ). The frequency response is a complex quantity having the magnitude response and the phase response ( Equation ). The frequency response exists if the system impulse response is abosolutely summable ( Equation ). In Example the impulse response is known and the problem is to evaluate the frequency response. Whereas in Example the frequency response is known and the problem is to find the impulse response for various cutoff frequencies (useful for chapter 5). In order to emphasized the small variations about the zero amplitude axis, the decibel (dB) scale is used instead of the linear scale ( equation ) , this makes the sidelobes more pronounced. When the input is the complex exponential e jωn MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadwgadaahaaWcbeqaaiaadQgacqaHjpWDcaWGUbaaaaaa@3AA9@ , the output is that particular input multiplied by the frequency response ( equation ). When several systems are in cascade, the overall response is the product of the individual responses. When several systems are in paralled, the overall response is the sum of the individual responses ( Figure ). When we know difference equation of a filter we can write the expression of its frequency response ( Equation ), and conversely. For this , three typical examples are given.

Introductory discrete Fourier transform (DFT) The discrete Fourier transform (DFT) relates discrete time to discrete frequency. It applies to any periodic signal, the same as the discrete-time Fourier series (DTFS) , as well as to any nonperiodic signal but imagined to repeat itself indefinitely. In the DFT the frequency is sampled to become discrete. Both the DFT ( Equation ) and the inverse DFT ( Equation ) are finite summations. A sequence of N samples is expanded into the same number N of spectral components , instead of an infinite series as the DTFS or a continuous integral as the DTFT. Hence the DFT is very computational efficient. The same as the CTFT and DTFT , the DFT applies to both signals and systems. There is the sampling theorem in the frequency domain similar to that in the time domain, this helps justifying the DFT. More of the DFT, and its computational algorithm FFT, will be discussed in a chapter 8.