Posts Tagged ‘Shannon’

Yesterday I visited a bookshop on Bangalore’s M.G. Road. I came across this classic – John G. Proakis, Digital Communitations, Fourth Edition, McGraw-Hill, 2001. I was overcome with a wave of nostalgia. This is a book I had used in engineering course in college some twelve years ago. I must have used the second edition. As a student, with limited financial resources, I had never bought a single technical book. The library supplied all that I needed. But yesterday I bought my first technical book. At $10.50, it was cheap by international standards and reasonably priced for India.

It has been many years since I read a book like this. The equations looked alien to me. I could not recognize the Fourier Transform. The complementary error function erfc(x) meant something but I had only a vague idea of its definition. The famous theorems of Shannon were somehow familiar but still a distant memory. Nonetheless, it was exciting to get back to the basics of digital communications.

In this post, I shall try to touch upon three things from this book and look at their relevance from the perspective of GSM/UMTS.

Noisy Channel Coding Theorem

This theorem from Shannon is stated as [1]:

There exist channel codes (and decoders) that make it possible to achieve reliable communication, with as small an error probability as desired, if the transmission rate R < C, where C is the channel capacity. If R > C, it is not possible to make the probability of error tend toward zero with any code.

A related equation to this defines the upper bound for the capacity C of a band-limited AWGN channel whose input is band-limited and power-limited [1]:

C = W log2 (1 + Pav/WNo) where

Pav is the average signal power
W is the signal bandwidth
No is the noise power spectral density

So for higher normalized capacity C/W (bits/s/Hz), power has to increase. So if we increase the modulation from QPSK to 16QAM as in HSDPA we get a higher normalized capacity. At the same time, we tighten the constellation if we keep the average signal power the same. A tighter constellation leads to higher BER. To obtain the same BER, we need to increase signal power. So although 16QAM gives us higher bit rate, this must be accompanied by higher power as indicated by the capacity formula. Another way to look at the same thing is to say that by moving to 16QAM we have more bits per symbol. Given that the energy per bit is the same, we require more power. This is understood by the following equation:

Pav/WNo = (Eb/No)(C/W) by substituting EbC for Pav.

On the other hand, for a fixed power, only an increase in bandwidth will give us higher capacity. This is true if we compare GSM against UMTS. The former has a bandwidth of only 200 kHz while the latter has one of 5 MHz.

A comparison of Shannon’s limit against the performance achieved by some standards is given in Figure 1 [2]. We can note that HSDPA is within 3 dB of the limit which is pretty good. We must remember that Shannon’s limit is for an AWGN channel, not for a fading channel (slow or fast). Yet fading characteristics can only make the performance worse, so that Shannon’s limit is still a definite upper bound on capacity. The fact that HSDPA can come within 3 dB of the limit is a tribute to advance receiver techniques. The Rake receiver is a key element in UMTS that makes the best of multipath diversity. Likewise, fast power control mitigates fades and improves BER performance. Without fast power control, the average SNR would be significantly higher to maintain the same BER.

Figure 1: Performance Relative to Shannon’s Limit
Performance against Shannon Limit

In a future post, I will look at MIMO in relation to Shannon’s limit.

Channel Coherence Bandwith

In a multipath channel, delayed versions of the signal arrive at the receiver with different delays, amplitudes and phases. A multipath intensity profile is the delay power spectrum of the channel in which most of the signal power arrives together with low delay and tapers off towards higher delay. The range of values over which this profile is non-zero is called the multipath spread or delay spread (Tm) of the channel. In practice, a certain percentage may be used to define the multipath spread, i.e. 95% of total power is within the multipath spread. In the frequency domain this can be shown to be related to coherence bandwith (Δf)c as

(Δf)c = 1/Tm

What this means is two frequencies separated by the coherence bandwith will fade differently through the channel. If a signal’s transmission bandwith is less than this amount, the channel is said to be frequency-nonselective or frequency flat. Otherwise, it is frequency-selective.

Multipath varies greatly in relation to the terrain. Table 1 prepared by the Institute of Technology Zurich is a nice summary of the range of values that multipath spread can take.

Table 1: Delay Spread for Different Terrains
Delay Spread Values

Apparent from this table are:

  • Urban areas have a smaller delay spread as compared to rural.
  • Indoor environments have small delay spreads.
  • Where there is a significant LOS path, delay spread is less.
  • There is some variability due to carrier frequencies and bandwith. In general, the terrain determines the delay spread.

Let us take 3 µs as the delay spread for an urban area. The coherence bandwidth would be 333 kHz. This is more than 200 kHz bandwith of GSM. Thus, this environment is frequency-flat for GSM. For WCDMA, the coherence bandwith is much smaller than 5 MHz. This is an important reason why WCDMA is called “wideband”. The channel is frequency-selective. Rake receivers are therefore quite important for WCDMA to reconstruct the signal without distortion.

For GSM, the fact that the channel is frequency-flat is used to good advantage. In other words, if the channel fades at one frequency it may not at another that’s at least one coherence bandwidth away. GSM employs frequency hopping which gives an improvement of 2-3 dB. On the other hand, if we consider a rural environment with delay spread of 0.03 µs, the coherence bandwith is 33 MHz which is quite high. GSM does not have frequencies spread over such a wide band. Thus frequency hopping is not likely to improve performance in such an environment, at least on this aspect. But frequency hopping does more than combat slow fading. It also mitigates interference which is why it is still useful when the delay spread is low.

Channel Coherence Time

For this we need to consider the time variations of the channel which is seen as a Doppler spread. Let us say that Sc(λ) is the power spectrum of the signal as a function of the Doppler frequency λ. The range over which this spectrum is non-zero is the Doppler spread Bd. Now we can define the coherence time (Δt)c as

(Δt)c = 1/Bd = co/fv where

co is the speed of light
f is the carrier frequency
v is the velocity of the receiver

A slowly changing channel has a large coherence time and a small Doppler spread. For example, in WCDMA if we consider a carrier frequency of 2 GHz, a vehicular speed of 250 kph, we get a coherence time of about 2.2 ms. This is smaller than UMTS 10 ms frame. Thus channel characteristics do change within a frame. This means that a UMTS frame experiences fast fading. How does the design combat fast fading? It does it simply by dividing the frame into 15 slots. Transmission power can be adjusted from one slot to the next. TPC bits sent in each frame enable this fast power control. Thus fast power control at the rate of 1.5 kHz help in combating fast fading. The net effect of this is that at the receiver, the required dBm is maintained to meet the target BLER set by outer loop power control.

In GSM, the same calculation yields a coherence time of 4.8 ms which is much larger than a GSM slot (0.577 ms). So GSM experiences slow fading. For GPRS on the other hand, multiple slots can be assigned to the MS. A frame is about 4.6 ms. So GPRS is on the border of slow to fast fading. This is tackled by transmitting an RLC/MAC block in four bursts over four frames.

Figure 2 is a representation of the time and frequency selective characteristics of a channel [4].

Figure 2: Time-and-Frequency Selective Channel
Time and Freq Selective Channel


  1. John G. Proakis, Digital Communitations, Fourth Edition, McGraw-Hill, 2001.
  2. Data Capabilities: GPRS to HSDPA and Beyond, Rysavy Research, 3G Americas, Sept 2005.
  3. Prof. Dr. H. Bölcskei, Physical Layer Parameters of Common Wireless Systems, Fundamentals ofWireless Communications, May 3, 2006.
  4. Klaus Witrisal, Mobile Radio Systems – Small-Scale Channel Modeling, Graz University of Technology, Nov 20, 2007.

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