1. Minjoong Rim, Dongguk University Signals and Systems 1 디지털통신 임 민 중 동국대학교 정보통신공학과

2. Minjoong Rim, Dongguk University Signals and Systems 2 Signals and Spectra

3. Minjoong Rim, Dongguk University Signals and Systems 3 Classification of Signals - 1 Deterministic signal there is no uncertainty with respect to its value at any time Random signal there is some degree of uncertainty before the signal actually occurs Periodic signal x(t) = x(t + T 0 ) for -  < t <  Nonperiodic(Aperiodic) signal Analog (continuous-time) signal continuous function of time, that is, uniquely defined for all t Discrete signal exists only at discrete times; characterized by a sequence of numbers defined for each time 101110 y = cos(2  f c t)

4. Minjoong Rim, Dongguk University Signals and Systems 4 Classification of Signals - 2 t t t t Continuous Non-periodic Continuous Periodic Discrete Non-periodic Discrete Periodic sampling decimation (down-sampling) interpolation (up-sampling) interpolation Classification of deterministic signalsContinuous to Discrete Discrete to Continuous Non-periodic to Periodic Sampling: the reduction of a continuous signal to a discrete signal Decimation: the processing of reducing the sampling rate of a signal Interpolation: a method of constructing new data points within the range of a discrete set of known data points

5. Minjoong Rim, Dongguk University Signals and Systems 5 Signal Representation s(t) T t s(-t) s(T-t) = s(-(t-T)) s(t-T) T t T t T t 0.5s(t) T t -s(t) s(t+T) T t T t -T t  T: s(T-T) = s(0) t  -T: s(-T+T) = s(0) t  -T: s(-(-T)) = s(T) t  0: s(T-0) = s(T) t  T: s(T-T) = s(0) t  0: -s(0) t  0: 0.5s(0) s(2t) T t T/2 t  T/2: s(2  T/2) = s(T)

6. Minjoong Rim, Dongguk University Signals and Systems 6 Special Functions - 1 Special Functions sinc function rectangular function triangular function impulse function (delta function) -area = 1 -amplitude =  -pulse width = 0 t 1 t 1/2 t 1 1 1 1 2345 t 1 sinc(0) = 1 sinc(n) = 0 n = 1,2,... -1,-2,...

7. Minjoong Rim, Dongguk University Signals and Systems 7 Special Functions - 2 Example: sinc  Amplitude Phase representing negative values sinc(0) = 1 sinc(n) = 0 n = 1,2,... -1,-2,...

8. Minjoong Rim, Dongguk University Signals and Systems 8 Circuits and Equipments - 1 ADC Analog-to-Digital Converter a device that converts a continuous physical quantity to a digital number that represents the quantity's amplitude DAC Digital-to-Analog Converter a device that converts digital data (usually binary) into an analog signal (current, voltage, or electric charge) Amp Amplifier an electronic device that increases the power of a signal Gain-controlled Amplifier Voltage-controlled amplifier an electronics amplifier that varies its gain depending on a control voltage discrete & quantized analog ADC digital analog DAC Amp Gain

9. Minjoong Rim, Dongguk University Signals and Systems 9 Circuits and Equipments - 2 Oscillator an electronic circuit that produces a repetitive, oscillating electronic signal, often a sine wave or a square wave Examples: signals broadcast by radio and television transmitters, clock signals that regulate computers and quartz clocks Voltage-controlled oscillator an electronic oscillator designed to be controlled in oscillation frequency by a voltage input Modulation Sinusoidal wave Power Amp Message Signal Carrier antenna baseband signal passband signal Oscillator Modulated Signal ppm: part per million (Example) 20ppm at 2GHz = 20  10 -6  2  10 9 Hz = 40000 Hz

10. Minjoong Rim, Dongguk University Signals and Systems 10 Linear Time-Invariant Systems Linear systems if x 1 (t)  y 1 (t) and x 2 (t)  y 2 (t) ax 1 (t) + bx 2 (t)  ay 1 (t) + by 2 (t) Time-invariant systems if x(t)  y(t) x(t -  )  y(t -  ) Causal No output prior to the time, t = 0, when the input is applied Linear network Input x(t) X(f) Output y(t) Y(f) h(t) H(f) input output linear time-invariant input output non-causal filter input  [ filter ]  filter output transmitted signal  [ communication channel ]  received signal

11. Minjoong Rim, Dongguk University Signals and Systems 11 Unit Impulse Function Unit impulse function an infinitely large amplitude pulse, with zero pulse width, and unity weight (area under the pulse), concentrated at the point where its argument is zero Characteristics  (t) = 0 for t  0  (t) is unbounded at t = 0 Impulse response the response when the input is equal to a unit impulse  (t)  (t)  h(t) Linear Time-Invariant System impulse impulse response t t t t continuous discrete input output

12. Minjoong Rim, Dongguk University Signals and Systems 12 Convolution - 1 Linear Time-Invariant System Convolution In linear time-invariant system, system output can be calculated with convolution of the input and the system impulse response Linear Time-Invariant System impulse impulse response Linear Time-Invariant System input (transmitted signal) output (received signal)  (t) h(t) x(t) y(t) = x(t)  h(t)

13. Minjoong Rim, Dongguk University Signals and Systems 13 Convolution - 2 Example 01 x(t) 1 01 h(t) 1 01 x(  )h(t-  ) tt-1 01 t 01 t 01 t Case 1 Case 2 Case 3 Case 4 01 y(t) 1 2     input system impulse response output tt t 0 < t-1 < 1 1 < t-1 h(t-  ) 0  h(-  ) tt-1  h(t-  )=h(-(  -t))

14. Minjoong Rim, Dongguk University Signals and Systems 14 Convolution - 3 Example 01 x(t) 1 0 2 h(t) 1 01 x(  )h(t-  ) tt-2 01 t 01 t 01 t Case 1 Case 2 Case 3 Case 4 01 t t-2 Case 5 01 y(t) 1 2 3    tt inputsystem impulse response output t t-2 < 0 and t < 1 0 < t-2 < 1 1 < t-2 h(t-  ) 0 -2  h(-  ) tt-2  h(t-  )=h(-(  -t))

15. Minjoong Rim, Dongguk University Signals and Systems 15 Discrete-Time Convolution - 1 Continuous-Time Convolution Discrete-Time Convolution x(t)h(t) tt inputsystem impulse response y (t) output x[n]h[n]y[n] inputsystem impulse responseoutput t

16. Minjoong Rim, Dongguk University Signals and Systems 16 Discrete-Time Convolution - 2 Discrete-time Convolution h[n]  x[n] h[-m] y[0] h[1-m] h[2-m] h[3-m] h[4-m] y[1] y[2] y[3] y[4] 0 0 123 1 0123 0 1 2 3 4

17. Minjoong Rim, Dongguk University Signals and Systems 17 Frequency - 1 Frequency

18. Minjoong Rim, Dongguk University Signals and Systems 18 Frequency - 2 Frequency and Wavelength = c / f 0 - : wave length -c: speed of light (= 3  10 8 m/s) -f 0 : frequency Example -f 0 = 1MHz  = 300m -f 0 = 1GHz  = 0.3m -f 0 = 2GHz  = ? Carrier a waveform (usually sinusoidal) that is modulated (modified) with an input signal for the purpose of conveying information f 0 = 1Hz  1 cycle/sec f 0 = 1KHz  1,000 cycle/sec f 0 = 1MHz  1,000,000 cycle/sec f 0 = 1GHz  1,000,000,000 cycle/sec

19. Minjoong Rim, Dongguk University Signals and Systems 19 Frequency - 3 AM/FM AM: 531~1,602 KHz FM: 88.1~107.9 MHz Digital TV 470~698 MHz ISM band Industrial, scientific, and medical radio bands -Microwave ovens, medical devices,... Despite the intent of the original allocations, short-range communications are widely used -WLAN, Bluetooth, near field communication,... 900 MHz, 2.4GHz, 5GHz Mobile Communication 800MHz, 1.8GHz, 2.1GHz, 2.3GHz 902 ~ 928 MHz (28MHz) 2.4 ~ 2.4835 GHz (83.5MHz) 5.725 ~ 5.825 GHz (125MHz) 800MHz (Cellular Band) SKT 2G, SKT 4G, LGU+ 4G 1.8GHz (PCS Band) KT 4G, SKT 4G, LGU+ 2G 2.1GHz (IMT2000 Band) SKT 3G, KT 3G 2.3GHz (WiBro) SKT WiBro, KT WiBro Licensed Band: Reserved for exclusive use of a single operator Deployed for single wireless technology Unlicensed Band: Open for anyone Deployed for any wireless technology as log as regulations are met

20. Minjoong Rim, Dongguk University Signals and Systems 20 Frequency Domain - 1 Frequency-domain Representation real imaginary Time DomainFrequency Domain f f0f0 f f Time Domain Frequency Domain

21. Minjoong Rim, Dongguk University Signals and Systems 21 Frequency Domain - 2 Example: Frequency-domain representation of cosine signal f f f Time DomainFrequency Domain real signal in the time domain minus frequency f0f0 2f 0 3f 0 minus frequency plus frequency 0.5 plus frequency

22. Minjoong Rim, Dongguk University Signals and Systems 22 Fourier Transform - 1 Fourier Transform for nonperiodic signal specifies the frequency-domain description or spectral content of the signal time domain to frequency domain Inverse Fourier Transform frequency domain to time domain Fourier Transform Inverse Fourier Transform t f frequency-domain time-domain energy signal

23. Minjoong Rim, Dongguk University Signals and Systems 23 Fourier Transform - 2 Fourier Transform of periodic signal Periodic signal can be represented as sum of complex exponentials with frequency nf 0 f time-domain frequency-domain = + + periodic discrete sum of complex exponentials with period T 0 / n (frequency = nf 0 ) T0T0 f0f0 2f 0 3f 0 f 0 =1/T 0 t Power signal: Note that only a single period is considered for integration

24. Minjoong Rim, Dongguk University Signals and Systems 24 Fourier Transform Properties - 1 Duality t f f t Note that the two equations are similar Time Domain Frequency Domain

25. Minjoong Rim, Dongguk University Signals and Systems 25 Fourier Transform Properties - 2 Scale Change Wide  Narrow Narrow  Wide Fast change in time-domain  wideband signals Time Domain Frequency Domain t f t f

26. Minjoong Rim, Dongguk University Signals and Systems 26 Fourier Transform Properties - 3 Discontinuous in time domain -High frequency component Smooth signal in time domain -mostly low frequency component t f t f Time Domain Frequency Domain abrupt change in time domain  wideband

27. Minjoong Rim, Dongguk University Signals and Systems 27 Fourier Transform Properties - 5 Minimum bandwidth -Large tail Excess bandwidth -Small tail f t f t Time Domain Frequency Domain

28. Minjoong Rim, Dongguk University Signals and Systems 28 Fourier Transform Properties - 6 Time Shift time delay  linear phase in the frequency domain ft f ft f 2X frequency → 2X phase delay 3X frequency → 3X phase delay

29. Minjoong Rim, Dongguk University Signals and Systems 29 Fourier Transform Properties - 7 Convolution Multiplication -Convolution in the time-domain transforms to multiplication in the frequency domain Frequency transfer function (frequency response)

30. Minjoong Rim, Dongguk University Signals and Systems 30 Fourier Transform Properties - 8 Modulation Time Domain Frequency Domain Multiplication  Convolution baseband bandpass or passband Message signal x Carrier f c = carrier frequency Message signal tf tf fcfc -f c tf fcfc ConvolutionMultiplication

31. Minjoong Rim, Dongguk University Signals and Systems 31 Filter - 1 Distortionless Transmission The output signal from an ideal transmission line may have some time delay compared to the input, and it may have a different amplitude than the input, but otherwise it must have no distortion Taking the Fourier transform System transfer function -Constant magnitude response -Phase shift must be linear with frequency time delay → linear phase f f t t distortionless transmission

32. Minjoong Rim, Dongguk University Signals and Systems 32 Filter - 2 Convolution in the time-domain transforms to multiplication in the frequency domain passbandstopband input system output

33. Minjoong Rim, Dongguk University Signals and Systems 33 Filter - 3 Ideal Low-Pass Filter Ideal Band-Pass Filter Ideal High-Pass Filter f f f f f f passband stopband passband stopband passband stopband bandwidth slope → delay Low pass filter passes low frequency components and stops high frequency components Filter output is a smooth version of the input signal t t High-pass Filter Low-pass Filter