Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes ” ) 5. Statistical Decision Theory - Hypothesis Testing (ch. 4 – part 0 “ Notes ” ) 2. Antipodal transmission (a special case of PAM) 3. Finite Energy Signal Space representations 4. Matched Filtering in AWGN for PAM antipodal links (ch. 2 – part 0 “Notes”) V4 – just moved to L2 the slides that we did not have time for

Review (and elaboration) of L1 (3 slides)

Target parameters for communication system design optimization Max Transmission rate (bit-rate, symbol-rate) Min Error Probability (bit/symbol/block error rate, outage) Min Power (PSD, SNR) Min Bandwidth (max spectral efficiency bps/Hz) Min Complexity (Cost) Min Delay (multi-user) Max # of users (multi-user) Min Mutual Interference Other parameters

Pulse Modulation and Pulse Amplitude Modulation (PAM) Figure 1.12: PULSE MODULATOR … … Index CODER (MAPPER) Bitstream

“Single t=0 – Isolated Pulse Amp. Mod. t t

End of Review

General PAM Link Analysis Figure 1.17: (“multi-shot” analysis in TA classes…) Channel Imp. Resp. PAM Pulse shape RX filter Imp. Resp. SAMPLER SLICER / DECISION Single-shot: Multi-shot: Single-shot:

Antipodal transmission system analysis

Antipodal digital link RX TX Medium Antipodal modulation: Special case of PAM modulation: modulating symbols equal +/-1

Example of antipodal Transmitter – flat pulses

Antipodal digital link analysis signal analysis: noise analysis: RX Medium TX TX+ Medium RX

Signal propagation through the receive filter done on the board

Noise propagation through the receive filter

Effective gaussian channel -axis

The statistics of decision DECIDE

Effective gaussian scalar channel Self-read

The gaussian-Q function

The Q-function is the complementary cdf of a normalized gaussian r.v. Figure 1.36:

The gaussian Q-function ^ Gaussian integral function or Q-function =Prob. of “upper tail” of normalized gaussian r.v.

Q(t) and some of its upper bounds Figure 1.38:

The dog wagging the (gaussian) tail vs. the tail wagging the dog

^ Deviation of t from the mean measured in units of the standard deviation ^^ ^ 0,1 Calculating cdf-s of gaussian variables with the Q-function ^^ done on the board Self-study

Error Probability calculation for the antipodal link

Probability of error (conditioned on the “0” hypothesis) noise indep. of signal

Probability of error (conditioned on the “1” hypothesis) Self-read

The two conditional Error Probabilities (graphical representation)

Total error probability (I) on the board

Total error probability (II) on the board

Total error probability (III) This completes the error prob. eval. for the antipodal link Next: optimize it, i.e. design the system to reduce BER SNR= antipodal system

Signal Spaces

Signal Spaces (vector spaces of functions of time) denotes a vector The key vector property: Vector Examples: (0) Arrows (i)D-tuples (ii)functions (iii)r.v.-s

VECTOR SPACES Self-study

Gram-Schmidt Self-study

Inner product spaces (I)

Inner product examples D

Norm, energy, distance The norm is the “length” of a vector or the root of its energy

Norm, energy – examples

Cauchy-Schwartz (C-S) inequality Example: geometric vectors:

C-S inequality – more examples

Correlation coefficient

Correlation coefficient - examples

…back to the error probability of the antipodal link

Total error probability (III) - revisit Rewrite in terms of vector notation for functions: | |

Error probability optimization – antipodal transmission For given h(t) maximize the s-factor by selecting the receive filter f(t): Use C-S ineq.: C-S ineq. becomes C-S eq. (i.e. s-factor is maximized) when:

The effect of scaling the receive and transmit filters (root) SNR does not change when both signal and noise are scaled by the same factor Constant gain (in RX) does not matter …but transmitting more power (making h(t) larger is beneficial – though we run into limits…

Matched filters

Matched filter f(t)=C h(-t) minimizes the error probability receive filter f(t) C

Optimal receiver for antipodal transmission is based on a matched receive filter Figure 1.48:

Matched filter f(t)=C h(-t) What is the value of the optimum (min.) error probability? It corresponds to the max s-factor: …by maximizing the SNR (or s-factor): minimizes the antipodal error probability…

Optimal Error Probability for antipodal link as a function of SNR

P is the average received power (energy per unit time) W is the bandwidth of the received signal (and the receive filter) A bound for the optimal probability of error in terms of bandwidth and received power Symbol rate cannot exceed twice the bandwidth Equality achieved for the so-called Nyquist pulses (see TA class) Self-study

Antipodal transmission operational point: For 10^-5 Error Probability, SNR must be 9.6 dB Figure 1.41:

Performance of antipodal receiver using a mismatched filter Figure 1.49: <1 Performance degraded Proof:

Causal Matched Filter When the input signal h(t) is causal, the impulse response h(-t) of the matched filter is non-causal. Sufficiently delaying this non-causal response turns it causal. Given the time-invariance, we must also delay our sampling instant (at time zero) Use this receiver front-end for optimal antipodal detection

Causal Matched Filter (II) Self-study

Correlators vs. matched filters

A correlator (I) (multiply & integrate implementation). implementation waveform in… …number out

A correlator (II) matched filter & sample implementation.

A correlator (III). Proof that “matched filter & sample” works as a correlator:

Correlators and their implementations Figure 1.52: Causal implementation Non-causal implementation The abstract view: May use any of these for optimal antipodal detection

Correlator optimization for maximum SNR the “Matched correlator” (I) f(t) = ?

What is SNR?

White Noise propagation through the Correlator Var =?

Correlator optimization for maximum SNR the “Matched correlator” (II)

Correlator optimization for maximum SNR the “Matched correlator” (III) Alternative “tricky” view: Maximize the output signal while constraining the kernel energy to be fixed But when the kernel energy is fixed so is the noise variance at the output! So signal is maximized, while the noise is fixed -> SNR is maximized It all happens when f(t) is “matched” to h(t) (any scale of f(t) will do) Self-study

Optimum Antipodal PAM Receiver: The matched correlator view Can use either the Multiply&integrate or Matched filter&sample implementations The SNR is maximized by the matched correlation receiver, yielding minimum BER Example: Integrate&dump receiver for flat pulse-shape g(t). Must multiply by a constant and integrate for T seconds, However multiplication is irrelevant (does not affect SNR) So, just integrate for T seconds (before dumping) RX Medium TX