Continuous – Time signals Two elements of signals are amplitude and time . Continuous-time (or analog) signals vary continuously with time. Waveforms are graphical illustration of signals. Signals are usually represented mathematically by expressions or equations. Sinusoidal signal (sinusoid) is very popular and has a very concise mathematical expression showing all parameters (amplitude, frequency,phase) ( Equation). Other waveforms, such as a square wave cannot be expressed mathematically so concisely (Equation). Signals can be deterministic, or random such as electric noise. Besides the sinusoid there are two special signals, namely the unit impulse (Equation) , the unit step (Equation). Signals can be real or complex. Concerning a complex signal x(t) we have magnitude (or modulus) denoted |x(t)| MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaacYhacaWG4bGaaiikaiaadshacaGGPaGaaiiFaaaa@3B53@ , phase (or phase angle) denoted Φ(t) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabfA6agjaacIcacaWG0bGaaiykaaaa@39D0@ or arg X(t) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadIfacaGGOaGaamiDaiaacMcaaaa@3933@ or ∠X(t) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabgcIiqlaadIfacaGGOaGaamiDaiaacMcaaaa@3AD1@ , complex conjugate x ∗ (t) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadIhadaahaaWcbeqaaiabgEHiQaaakiaacIcacaWG0bGaaiykaaaa@3A79@ (Equation). Complex exponential (Equation) is a good representation of complex signals, it can be considered as a phasor (rotating vector). We should know how to obtain the real part from a complex exponential (Equation) and (Equation).

Noise There exist several types of noise, the most popular is thermal noise. Noise can be internal or external (interference). White noise has its power spectral density unchanged with frequency. When a white noise passes through a filter, the output will be no longer white. The probability of occurrence at different amplitudes of noise is very important in analysis, this concerms random variable, probability density function (PDF), and cumulative distribution function (CDF). The uniform and gaussian distributions are usually considered. The two main parameters of distributions are mean and variance.

Signal sampling Sampling a continuous-time signal turns it into a discrete-time one. In most case we use uniform sampling. Sampling is a multiplying proces, implemented, in principle, just by a switch (Fig). We call T s MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadsfadaWgaaWcbaGaam4Caaqabaaaaa@3801@ the sampling interval (or sampling period), and f s =1/ T s MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadAgadaWgaaWcbaGaam4CaaqabaGccqGH9aqpcaaIXaGaai4laiaadsfadaWgaaWcbaGaam4Caaqabaaaaa@3C8E@ the sampling frequency (or sampling rate). Time index n is an integer number. For analog signals, we denote by F x MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadAeadaWgaaWcbaGaamiEaaqabaaaaa@37F8@ and T x MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadsfadaWgaaWcbaGaamiEaaqabaaaaa@3806@ their frequency and period , resprectively. In order the samples can represent correctly the original analog signal, the sampling frequency must be greater than twice its maximum frequency component ( f s >2 F M MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadAgadaWgaaWcbaGaam4CaaqabaGccqGH+aGpcaaIYaGaamOramaaBaaaleaacaWGnbaabeaaaaa@3BAA@ (Equation). This is the sampling theorem. Nyquist rate is 2 F M MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaaikdacaWGgbWaaSbaaSqaaiaad2eaaeqaaaaa@3889@ and Nyquist interval is [− f s /2, f s /2] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaacUfacqGHsislcaWGMbWaaSbaaSqaaiaadohaaeqaaOGaai4laiaaikdacaGGSaGaaGjbVlaadAgadaWgaaWcbaGaam4CaaqabaGccaGGVaGaaGOmaiaac2faaaa@41FE@ . When a signal is sampled below the Nyquist rate the aliasing occurs , which should be avoided in most cases, by the use of an antialiasing prefilter or by raising the sampling frequency to satisfy the sampling theorem. Section shows us how to find the alias (aliased signal).

Discrete-time signals Discrete-time signals must be quantized and binary encoded to become binary digital signals, but these two processes are usually understood, hence discrete-time signals and digital signals usually mean the same thing . Smilarly , we don’t usually differentiate discrete-time signal processing (DTSP) and digital signal processing (DSP) . Example gives a general DSP system. Discrete-time signals may be infinite duration or finite daration. A discrete-time signal is just a sequence of numbers (real or complex) and can be written as such. Basic digital signals are unit sample or unit impulse (Equation), unit step (Equation), unit ramp (Equation), real exponential (Equation), and complex exponential (Equation and Equation). When comparing an analog consinusoid to the corresponding digital consinusoid we obtain the useful relations ω=Ω T s =2π F f s and f= F f s MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabeM8a3jabg2da9iabfM6axjaadsfadaWgaaWcbaGaam4CaaqabaGccqGH9aqpcaaIYaGaeqiWda3aaSaaaeaacaWGgbaabaGaamOzamaaBaaaleaacaWGZbaabeaaaaGccaaMf8Uaamyyaiaad6gacaWGKbGaaGzbVlaadAgacqGH9aqpdaWcaaqaaiaadAeaaeaacaWGMbWaaSbaaSqaaiaadohaaeqaaaaaaaa@4D98@ where ω MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabeM8a3baa@37D1@ is the digital angular frequency (radians/sample), ω=2πf MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabeM8a3jabg2da9iaaikdacqaHapaCcaWGMbaaaa@3C3B@ with f is the digital frequency (cycles/sample), Ω MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabfM6axbaa@3792@ the analog angular frequency (radians/sec), Ω=2πF MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabfM6axjabg2da9iaaikdacqaHapaCcaWGgbaaaa@3BDC@ with F the analog frequency (Hertz or cycles/sec), and f s MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadAgadaWgaaWcbaGaam4Caaqabaaaaa@3813@ is the sampling frequency (Hertz or samples/sec). A sampled analog simusoidal signal may be periodic or not (Equation).

Discrete-time systems Discrete-time (or digital) systems process the input signal x(n) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadIhacaGGOaGaamOBaiaacMcaaaa@394D@ to give an output signal y(n) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadMhacaGGOaGaamOBaiaacMcaaaa@394E@ differed from the input in some aspect (amplitude, frequency, phase…) . System is described (or characterized) by its input-output signal difference equation. Especially, y(n)=x(n−1) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadMhacaGGOaGaamOBaiaacMcacqGH9aqpcaWG4bGaaiikaiaad6gacqGHsislcaaIXaGaaiykaaaa@3F45@ represents an unit delay, and y(n)=x(n+1) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadMhacaGGOaGaamOBaiaacMcacqGH9aqpcaWG4bGaaiikaiaad6gacqGHRaWkcaaIXaGaaiykaaaa@3F3A@ an unit advance. Section gives the basic building blocks of discrete-time systems comprising adder (summer), subtractor, scalar multiplier, signal multiplier, signal squaring, unit delay, and unit advance. From a given input-output difference equation of a system we can build its structure using the appropriate basic building blocks, and vice versa. Systems can be nonrecursive (feedforward) or recursive (with feedback).

Types of discrete-time systems First we have memoryless systems and systems with memory. Next come causal and noncausal systems. Real-time precessing systems must be causal. Time-invariant systems (Fig) are easier to analyze than time-variant systems. Linearity of systems (Fig) includes both proportionality an superprosition. Unless ortherwise specified, we assume only linear and time-invariant (LTI) systems . Time-invariant also means shift-invariant, hence the name linear and shift-invariant (LSI) systems.