Lecture Notes: EEEC6440315 Communication Systems - Inter Symbol Interference and Pulse Shaping
Download as PPTX, PDF0 likes1,855 views
This document discusses inter-symbol interference (ISI) that occurs when pulses transmitted through a band-limited channel spread into adjacent time slots, and various pulse shaping techniques to eliminate ISI. It explains that rectangular pulses cause ISI in practical band-limited channels, and introduces Nyquist's criterion for zero-ISI transmission. The document also describes raised cosine pulse shaping, which is commonly used when the symbol rate is less than the Nyquist rate, and provides an example of its use in WCDMA cellular systems.
Lecture Notes: EEEC6440315 Communication Systems - Inter Symbol Interference and Pulse Shaping
- 1. EEEC6440315 COMMUNICATION SYSTEMS Inter Symbol Interference And Pulse Shaping FACULTY OF ENGINEERING AND COMPUTER TECHNOLOGY BENG (HONS) IN ELECTRICALAND ELECTRONIC ENGINEERING Ravandran Muttiah BEng (Hons) MSc MIET
- 2. Band Limited Channel And Inter Symbol Interference Rectangular pulses are suitable for infinite bandwidth channels (practically - wideband). Practical channels are band limited ⇒ Pulses spread in time and are smeared into adjacent slots. This is Inter Symbol Interference (ISI) 1
- 3. 2 1 0 0 0 1 0 1 1 0 0 0 0 0 0 t t t t t t Intersymbol Interference Sampling Points (Transmitter Clock) Sampling Points (Receiver Clock) Sampling Points (Receiver Clock) Input Binary Waveform Individual Pulse Response Received Waveform Figure 1: Inter Symbol Interference. 𝑇s 𝑇s 𝑇s
- 4. 3 Figure 2: Channel noise. Eye Diagram Convenient way to observe the effect of ISI and channel noise on an oscilloscope. Best Sampling Time 0 1 1 0 0 1 Eye Pattern Waveform Ideal Filtering Filtering With ISI Noise Plus ISI 𝑇b 𝑇b Maximum Distortion Noise Margin t t t
- 5. 4 Peak distortion Noise Margin Distortion of Zero Crossings Sensitivity to Timing Error Optimum Sampling Time Figure 3: Eye diagram. Oscilloscope presentations of a signal with multiple sweeps (triggered by a clock signal), each is slightly larger than symbol interval. Quality of a received signal may be estimated. Normal operating conditions (no ISI, no noise) ⇒ Eye is open. Large ISI or noise ⇒ Eye is closed.
- 6. 5 Timing error allowed ⇒ width of the eye, called eye opening (preferred sampling time – at the largest vertical eye opening). Sensitivity to timing error ⇒ Slope of the open eye evaluated at the zero crossing point. Noise margin ⇒ The height of the eye opening.
- 7. 6 Transmitting Filter 𝐻T 𝑓 Channel (Filter) Characteristics 𝐻C 𝑓 Receiver Filter 𝐻R 𝑓 Transmission Over A Band - Limited Channel Flat Top Pulses Recovered Rounded Pulse (to sampling and decoding circuits) 𝑥 𝑡 y 𝑡 Figure 4: Baseband transmission system. Input PAM signal, 𝑥 𝑡 = 𝑛 𝑎𝑛𝑠T 𝑡 − 𝑛𝑇 Output signal, y 𝑡 = 𝑛 𝑎𝑛𝑠 𝑡 − 𝑛𝑇 , 𝑠 𝑡 = 𝑠T 𝑡 ∗ ℎT 𝑡 ∗ ℎC 𝑡 ∗ ℎR 𝑡 = 𝑠T 𝑡 ∗ ℎ 𝑡 Sampled output at 𝑡 = 𝑚𝑇, 𝑦𝑚 = 𝑛 𝑎𝑛𝑠𝑚−𝑛 = 𝑎𝑚𝑠0 + 𝑛≠𝑚 𝑎𝑛𝑠𝑚−𝑛 = 𝑦 𝑚𝑇 𝑠𝑚−𝑛 = 𝑠 𝑚𝑇 − 𝑛𝑇 transmitted symbol inter symbol interference
- 8. 7 Pulse Shaping To Eliminate ISI In 1928 Nyquist discovered 3 methods to eliminate ISI: • Zero ISI pulse shaping. • Controlled ISI (Eliminated later on by an Equalizer). • Zero average ISI (negative and positive areas under the pulse in an adjacent interval are equal). • Zero ISI pulse shaping, 𝑠 𝑛𝑇 = 𝑠0 𝑛 = 0 0 𝑛 ≠ 0 𝑦𝑚 = 𝑎𝑚𝑠0 + 𝑛≠𝑚 𝑎𝑛𝑠𝑚−𝑛 ⇒ 𝑦𝑚 = 𝑎𝑚𝑠0
- 9. 8 Example 1: 𝑠 𝑡 = sinc 𝑓0𝑡 𝑠 𝑛𝑇 = sinc 𝑛 = 0, 𝑛 ≠ 0 Hence, 𝑠𝑖𝑛c pulses allows to eliminate ISI at sampling instants. Figure 5: Zero ISI 𝑠𝑖𝑛c pulse. -3 -2 -1 -0 1 2 3 -0.3 0 0.3 0.6 0.9
- 10. 9 Nyquist Criterion For Zero ISI Provides a generic solution to the zero ISI problem. A necessary and sufficient condition for s 𝑡 to satisfy, 𝑠 𝑛𝑇 = 𝑠0 𝑛 = 0 0 𝑛 ≠ 0 ⇔ 𝑚=−∞ ∞ 𝑆s 𝑓 + 𝑚 𝑇 = 𝑇 Consider a channel band-limited to 0, 𝐹max for 3 cases. • 1 𝑇 = 𝑓0 > 2𝐹max ⇒ No way to eliminate ISI. • 𝑓0 = 2𝐹max ⇒ Only sinc 𝑓0𝑡 eliminates ISI. The highest possible transmission rate 𝑓0 for transmission with zero ISI is 2𝐹max, not 𝐹max. • 𝑓0 < 2𝐹max ⇒ Many signals may eliminated ISI in this case. Note: 𝑓0 = 1 𝑇 is the transmission rate (symbols/second)
- 11. 10 Figure 6: 𝑓0 > 2𝐹max, 𝑓0 = 2𝐹max and 𝑓0 < 2𝐹max. 1 𝑇 = 𝑓0 > 2𝐹 max 𝑓0 = 2𝐹 max 𝑓0 < 2𝐹 max 𝑚=−∞ ∞ 𝑆s 𝑓 + 𝑚 𝑇 = 𝑍 𝑓 𝑚=−∞ ∞ 𝑆s 𝑓 + 𝑚 𝑇 = 𝑍 𝑓 𝑚=−∞ ∞ 𝑆s 𝑓 + 𝑚 𝑇 = 𝑍 𝑓 𝑊 = 𝐹 max − 1 𝑇 − 1 𝑇 + 𝑊 1 𝑇 − 𝑊 −𝑊 0 0 𝑊 1 𝑇 1 𝑇 + 𝑊 − 1 𝑇 1 𝑇 𝑊 = 1 2𝑇 − 1 𝑇 1 𝑇 − 1 𝑇 + 𝑊 −𝑊 𝑊 1 𝑇 − 𝑊 Case 1: Case 2: Case 3: ⋯ ⋯ ⋯ ⋯
- 12. 11 Raised Cosine Pulse When 𝑓0 < 2𝐹max, raised cosine pulse is widely used. The raised cosine pulse has Fourier transform, which its spectrum is, 𝐻 𝑓 = 𝑇𝜑, 0 ≤ 𝑓 ≤ 1−𝛼 2𝑇 𝑇 2 1 + cos π𝑇 𝛼 𝑓 − 1−𝛼 2𝑇 , 1−𝛼 2𝑇 ≤ 𝑓 ≤ 1+𝛼 2𝑇 0, 𝑓 > 1+𝛼 2𝑇 where 𝛼 is roll-off factor, 0 ≤ 𝛼 ≤ 1. The bandwidth above the Nyquist frequency 𝑓0 2 = 1 2𝑇 is called excess bandwidth. Where, 𝑓0 = 1 𝑇 Time-domain waveform of the raised cosine pulse, ℎ 𝑡 = sin π𝑡 𝑇 π𝑡 𝑇 cos 𝛼π𝑡 𝑇 1− 2𝛼𝑡 𝑇 2
- 13. 12 When 𝛼 = 0, pulse reduces to sinc 𝑡𝑓0 = sinc 𝑡 𝑇 and 𝑓0 = 2𝐹max. 𝛼 = 1, symbol rate 𝑓0 = 𝐹max. Tails decay as 𝑡3 , mistiming is not a big problem. Smooth shape of the spectrum, easier to design filters. 1 0.5 0 0 0.5 1 1 0 -1 -2 -3 Figure 7: Raised Cosine Spectrum. Figure 8: Time Domain Pulse. 𝛼 = 1 𝛼 = 0.5 𝛼 = 0 -0.5 -1 0 0.5 1 2 3
- 14. 13 Example 2: Wideband Code-Division Multiple-Access (WCDMA) is the main technology behind 3rd Generation (3G) cellular systems implements a Square-Root Raised Cosine pulse shape with excess bandwidth 𝛼 = 0.22, which translates the signaling rate of 3.84 MHz to a bandwidth of 0.5 × 1 + 𝛼 × 2 × 3.84 = 4.68 MHz (the factor 2 arises for the RF bandwidth). Accounting for the guard-bands to minimise interference between neighboring channels, the signal bandwidth in WCDMA systems is 5 MHz.
- 15. (1) Leon W. Couch II, Digital and Analogue Communication Systems, Prentice Hall, 1999. References 14