EE360 – Lecture 2 Outline Announcements: Class mailing list: (subscribe Wireless network lunches: to subscribe. Website is wsl.stanford.edu/projects/snrc.html. First lunch this quarter is April 16, noon, in CIS Broadcast Channels TD, FD, CD: Practical Implications Capacity of Broadcast Channels with AWGN Broadcast Channels with ISI Fading Broadcast Channels

Broadcast Channels Synchronization easy. Interference signals follow same path as desired signal (no near-far problem) Complexity/power at transmitter less restricted than at receiver.

Frequency Division Advantages Narrowband channels (no ISI) Low complexity Allows cts. time transmission and channel estimation. Disadvantages Radios must be frequency-agile Handoff complicated by continuous transmission Dedicated channels (idle ones wasted) Difficult to allocate multiple channels per user Total system bandwidth divided into orthogonal channels assigned to different users. FD alone not used in current digital systems

Time Division Advantages No need for frequency agility Discontinuous transmission facilitates handoff and reduces power consumption. Easy to allocate multiple channels/user Disadvantages Synchronization required Multipath destroys slot orthogonality Typically requires ISI mitigation (short timeslots) Idle channels may be wasted Short transmissions make equalization and dynamic resource allocation hard. Time divided into orthogonal slots, with different timeslots assigned to different users. TD used (with frequency hopping) in GSM

Code Division Advantages With semi-orthogonal codes, no hard limit to # of users in system (soft capacity - system is interference-limited) Interference reduction techniques increase capacity Synchronization not required Can allocated multiple “channels”/user using multicode or multirate techniques. No near-far problem on downlink Disadvantages Complexity Multipath creates multiple interferers Orthogonal or semi-orthogonal codes used to modulate each users signal. Code properties used to separate users at the receiver. CD used in IS-95 and many 3G propositions

Examples AMPS FDMA/FDD GSM (EDGE)TDMA/FDD IS-54 and IS-136TDMA/FDD JDCTDMA/FDD IS-95CDMA/FDD IMT-2000CDMA/FDD

Broadcast Channel Capacity Region in AWGN Model One transmitter, two receivers with spectral noise density n 1, n 2 : n 1 <n 2. Transmitter has average power S and total bandwidth B. Single User Capacity: Maximum achievable rate with asymptotically small P e Set of achievable rates includes (C 1,0) and (0,C 2 ), obtained by allocating all resources to one user.

Rate Region: Time Division Time Division (Constant Power) Fraction of time  allocated to each user is varied Time Division (Variable Power) Fraction of time  and power  i  allocated to each user is varied

Rate Region: Frequency Division Frequency Division Bandwidth B i and power S i allocated to each user is varied. Equivalent to TD for B i =  i B and S  =  i  i.

Superposition Coding Best user decodes fine points Worse user decodes coarse points

Code Division Superposition Coding Coding strategy allows better user to cancel out interference from worse user. DS spread spectrum with spreading gain G and cross correlation  12 =   =G: By concavity of the log function, G=1 maximizes the rate region. DS without interference cancellation

Broadcast Channels with ISI ISI introduces memory into the channel What strategy achieves capacity region of a broadcast channel with ISI and correlated noise? The coding strategy decomposes the channel into parallel degraded broadcast channels: Superposition coding is applied to each subchannel. Power must be optimized across subchannels and between users in each subchannel.

Broadcast Channel Model Both H 1 and H 2 are finite IR filters of length m. The w 1k and w 2k are correlated noise samples. For 1<k<n, we call this channel the n-block discrete Gaussian broadcast channel (n-DGBC). The channel capacity region is C=(R 0,R 1,R 2 ). w 1k H1()H1()H2()H2() w 2k xkxk

Circular Channel Model For a FIR channel {h i : i=1,…m} and n>m define Applying this to {h 1i } and {h 2i } yields the n-Block Circular Gaussian Broadcast Channel (n-CGBC) 0<k<n where ((. )) denotes addition modulo n.

Equivalent Channel Model Taking DFTs of both sides yields Dividing by H and using additional properties of the DFT yields 0<j<n ~ where {V 1j } and {V 2j } are independent zero-mean Gaussian random variables with 0<j<n

Channel Decomposition The n-CGBC thus decomposes to a set of n parallel discrete memoryless degraded broadcast channels with AWGN. Can show that as n goes to infinity, the circular and original channel have the same capacity region The capacity region of parallel degraded broadcast channels was obtained by El-Gamal (1980) The power constraint on the original channel is converted by Parseval’s theorem to on the equivalent channel.

Capacity Region of Parallel Set Achievable Rates (no common information) Capacity Region For 0<  < find {  j }, {P j } to maximize R 1 +  R 2 +  P j. Let (R 1 *,R 2 * ) n,  denote the corresponding rate pair. C n ={(R 1 *,R 2 * ) n,  : 0<  < }, C =liminf n C n.

Limiting Capacity Region