N - Channel Allocation

N - Channel Allocation
Time Limit:1000MS     Memory Limit:10000KB     64bit IO Format:%I64d & %I64u

Description

When a radio station is broadcasting over a very large area, repeaters are used to retransmit the signal so that every receiver has a strong signal. However, the channels used by each repeater must be carefully chosen so that nearby repeaters do not interfere with one another. This condition is satisfied if adjacent repeaters use different channels. 

Since the radio frequency spectrum is a precious resource, the number of channels required by a given network of repeaters should be minimised. You have to write a program that reads in a description of a repeater network and determines the minimum number of channels required.

Input

The input consists of a number of maps of repeater networks. Each map begins with a line containing the number of repeaters. This is between 1 and 26, and the repeaters are referred to by consecutive upper-case letters of the alphabet starting with A. For example, ten repeaters would have the names A,B,C,...,I and J. A network with zero repeaters indicates the end of input. 

Following the number of repeaters is a list of adjacency relationships. Each line has the form: 

A:BCDH 

which indicates that the repeaters B, C, D and H are adjacent to the repeater A. The first line describes those adjacent to repeater A, the second those adjacent to B, and so on for all of the repeaters. If a repeater is not adjacent to any other, its line has the form 

A: 

The repeaters are listed in alphabetical order. 

Note that the adjacency is a symmetric relationship; if A is adjacent to B, then B is necessarily adjacent to A. Also, since the repeaters lie in a plane, the graph formed by connecting adjacent repeaters does not have any line segments that cross. 

Output

For each map (except the final one with no repeaters), print a line containing the minumum number of channels needed so that no adjacent channels interfere. The sample output shows the format of this line. Take care that channels is in the singular form when only one channel is required.

Sample Input

2
A:
B:
4
A:BC
B:ACD
C:ABD
D:BC
4
A:BCD
B:ACD
C:ABD
D:ABC
0

Sample Output

1 channel needed.
3 channels needed.
4 channels needed. 

### OFDM Reference Signal in Wireless Communication In wireless communication systems, Orthogonal Frequency Division Multiplexing (OFDM) uses reference signals to assist with various tasks such as channel estimation, synchronization, and equalization. These reference signals are also known as pilot symbols or training sequences. #### Purpose of OFDM Reference Signals Reference signals play an essential role in ensuring reliable data transmission over multipath fading channels. They help estimate the time-varying characteristics of the propagation environment so that compensation can be applied at the receiver side[^1]. The primary purposes include: - **Channel Estimation**: To accurately determine the complex gains experienced by each subcarrier. - **Synchronization**: For maintaining symbol timing alignment between transmitter and receiver. - **Phase Noise Compensation**: Mitigating phase noise effects introduced during modulation/demodulation processes. #### Design Considerations for OFDM Pilots When designing these pilots within an OFDM frame structure, several factors must be considered: - **Placement Strategy**: Deciding whether to place them uniformly across all frequency bins or adopt more sophisticated patterns like comb-type arrangements depends on specific application requirements [^4]. ```matlab % Example MATLAB code snippet showing placement strategy NFFT = 64; % FFT size numPilotsPerSymbol = floor(NFFT/8); % Number of pilots per OFDM symbol pilotIndices = linspace(1,NFFT,numPilotsPerSymbol); ``` - **Power Allocation**: Ensuring sufficient energy allocation without compromising payload capacity remains crucial since higher-powered pilots improve accuracy but reduce spectral efficiency [^2]. - **Orthogonality Maintenance**: Maintaining orthogonality among different users sharing same resources prevents interference which could degrade overall performance metrics significantly [^3]. #### Mathematical Representation Let us denote \( y[n] \) as received samples after passing through multi-path fading channel represented mathematically using impulse response coefficients h[m]: \[ y[n]=\sum _{m=0}^{L-1}\alpha _me^{-j2\pi f_nmT_s}x[n-m]+w[n],n=0,\ldots ,N-1 \] Where L represents maximum number of taps present inside this linear filter model while w[] signifies additive white Gaussian noise component affecting every sample independently according to its variance σ² . Here α_m corresponds to attenuation factor associated ith tap alongwith corresponding Doppler shift term exp(-j*2πf_n*mTs)[^5]. ```python import numpy as np def generate_pilot_sequence(length): """Generate random QPSK-based pilot sequence.""" return np.random.choice([1+1j,-1+1j,1-1j,-1-1j], length) pilots = generate_pilot_sequence(numPilotsPerSymbol) print(pilots) ```
评论
添加红包

请填写红包祝福语或标题

红包个数最小为10个

红包金额最低5元

当前余额3.43前往充值 >
需支付:10.00
成就一亿技术人!
领取后你会自动成为博主和红包主的粉丝 规则
hope_wisdom
发出的红包
实付
使用余额支付
点击重新获取
扫码支付
钱包余额 0

抵扣说明:

1.余额是钱包充值的虚拟货币,按照1:1的比例进行支付金额的抵扣。
2.余额无法直接购买下载,可以购买VIP、付费专栏及课程。

余额充值