Dept. of EE, NDHU 1 Chapter One Signals and Spectra

Dept. of EE, NDHU 2 Why Digital ? Advantages –Digital signals are more easily regenerated –Digital circuits are more reliable and can be produced at lower cost –Different types of digital signals can be treated as identical signals in transmission and switching –Digital techniques are naturally to signal processing functions that protect against interference and jamming, or provide encryption Costs –Very signal-processing intensive –Need to synchronize at various levels –Non-graceful degradation

Dept. of EE, NDHU 3 Pulse Degradation and Regeneration

Dept. of EE, NDHU 4 Typical Digital Communication System

Dept. of EE, NDHU 5 Digital Communication Transformations Formatting –Analog source: audio, speech, video signal –Digital source: computer data, digital image –Convert the source into a sequence of binary sequence Source encoding –Efficiently convert the digital symbol into a sequence of binary digits –Data compression: MEG encode, JPEG, Huffiman coding, MP3 Channel encoder –Introduce some redundancy in the binary information sequence that can be used at the receiver to overcome the effects of noise and encounter the channel

Dept. of EE, NDHU 6 Digital Communication Transformations Pulse modulation –Map the binary information sequence into signal waveform Bandpass signaling –Coherent: PSK, FSK, GMSK –Non-coherent: DPSK, FSK

Dept. of EE, NDHU 7 Basic Digital Communication Nomenclature (Textual messages) (Characters) (7-bit ASCII) (Symbol) (Bandpass digital waveform)

Dept. of EE, NDHU 8 Performance Criteria Analog communication systems –The figure of merit is a fidelity criterion –For example signal-to noise ratio, percent distortion, or expected mean-square error between the transmitted and received waveforms Digital communication systems –Probability of incorrectly detecting a digit, or P E

Dept. of EE, NDHU 9 Classification of Signals Deterministic and Random signals –Deterministic signal means that there is no uncertainty with respect to its value at any time, for example x(t)=5 cos 10t –Random signal means that there is some degree of uncertainty before signal actually occurs –Random waveform is NOT possible to write an explicit expression, can be described by probabilities and statistical averages Periodic and Non-periodic signals –A signal x(t) is periodic in time if there exits a constant T 0 such that –No value of T 0 that satisfies equation (1.2) is called non-periodic signal

Dept. of EE, NDHU 10 Classification of Signals Analog and Discrete signals –x(t) and x(kT) Energy and Power signals –Energy signal is defined by the signal has nonzero but finite energy for all time –Power signal is defined by the signal has finite but nonzero power for all the time –Periodic signal and random signal are generally classified as power signals –Both deterministic and non-periodic signals are generally classified as energy signals

Dept. of EE, NDHU 11 Spectral Density Energy spectral density –Where is defined as energy spectral density (ESD) of the signal x(t) Power spectral density –The power spectral density (PSD) is –See Example 1.1

Dept. of EE, NDHU 12 Autocorrelation A measure of how closely the signal matches a copy of itself as the copy is shifted  in the time

Dept. of EE, NDHU 13 Random Process

Dept. of EE, NDHU 14 Random Process Stationary –Strict-sense stationary if none of statistics are affected by a shift in the time origin –Wide-sense stationary if Ergodic –Time averages equal ensemble averages –For example, –The statistical properties of the process can be determined by time averaging over a single sample function

Dept. of EE, NDHU 15 Some Useful Probability Distributions Binormial Distribution –Let X be a discrete random variable X=1 or X=0, with probability p an 1-p Uniform Distribution Gaussian (normal) Distribution Chi-square (exponential) Distribution Rayleigh Distribution Ricean Distribution Lognormal Distribution

Dept. of EE, NDHU 16 Autocorrelation and Power Spectral Density

Dept. of EE, NDHU 17 Autocorrelation and Power Spectral Density

Dept. of EE, NDHU 18 Normalized Gaussian Probability Density Function

Dept. of EE, NDHU 19 White Noise Figure 1.8 (a) Power spectral density of white noise.(b) Autocorrelation function of white noise.

Dept. of EE, NDHU 20 Linear Systems Frequency response Power spectral density Distortionless transmission

Dept. of EE, NDHU 21 Ideal Filter Transfer function Impulse response

Dept. of EE, NDHU 22 Impulse Response of the Ideal Low-pass Filter

Dept. of EE, NDHU 23 Realizable Filter

Dept. of EE, NDHU 24 Butterworth Filter Magnitude frequency response for the n-th order

Dept. of EE, NDHU 25 RC Filtering an Ideal Pulse

Dept. of EE, NDHU 26 Baseband versus Bandpass

Dept. of EE, NDHU 27 Bandwidth Dilemma Strictly bandlimited signal Strictly time limited signal For all bandlimited spectra, the waveform are not realizable, and for all realizable waveforms, the absolute bandwidth is infinite.

Dept. of EE, NDHU 28 Bandwidth Criteria Fig. Bandwidth of digital data. (a) Half-power. (b) Noise equivalent. (c) Null to null. (d) 99% of power. (e) Bounded PSD (defines attentuation outside bandwidth) at 35 and 50 dB.