The subchapter treats the problem of minimizing the intersymbol interference;  presents the frequency domain  requirements for no intersymbol interference,  the time response of a transmission system corresponding to no intersymbol interference, employs as  analytical tool the sampling theorem to determine the frequency characteristic for no intersymbol interference introducing terms as "Nyquist interval", "Nyquist frequency" and "Nyquist characteristic", presents a  class of Nyquist characteristics extensively used and studied - "raised cosine".

CONTENT

Notice from Eq. 2.4 that intersymbol interference can only be eliminated by making x_n =0 for all n¹ 0. In other words, the requirement is that precursors and tails of x(t) pass through 0 at regular T - sec. intervals. An example of such pulse is shown in Figure 2.11. It should be clear that these pulses could be amplitude modulated and transmitted at T - sec. intervals without overlap at the sampling instants.

But in design and analysis of baseband transmission system it is convenient, and often necessary, to be able to specify in the frequency domain the requirements for no intersymbol interference. So, the problem is how X(w ), the Fourier transform of x(t), should be to have x_n=0 for n¹ 0.

More generally, our problem is to specify the Fourier transform X(w ) of a time function x(t) when the samples x_n=x(nT) are given. The fundamental analytical tool, which we shall employ for this purpose, is the sampling theorem. The theorem enables us to express the time response x(t) and the frequency response X(w ) for a function band-limited to [-f_Max, f_Max ] in terms of samples taken at 1/2f_Max - sec intervals. The interval 1/2f_Max sec is known as the Nyquist interval or, alternatively, the frequency 1/2T Hz for sampling interval of T sec is known as the Nyquist frequency. The point of the theorem is that a function band-limited to the frequency range [-f_Max, f_Max] Hz has exactly 2f_Max degrees of freedom per second. When these are specified, the function is known exactly.

In baseband pulse transmission we are concerned with samples of x(t) taken at T - sec intervals. If X(w ) is band-limited to the Nyquist frequency f_N=1/2T Hz, these samples uniquely determine the function x(t). If X(w) is band-limited to a frequency smaller than f_N there isn’t a function x(t), and implicitly an X(w ), corresponding to an imposed set of samples x(nT).
If X(w ) is band-limited to some frequency higher than f_N, there exist an infinity of functions x(t), and the corresponding
X(w ), having the same samples sequence { x_n}. All these characteristics X(w ), corresponding to the same samples sequence { x_n}, are equivalent. The characteristic band-limited to the Nyquist band and corresponding to the samples sequence { x_n}is called the equivalent Nyquist characteristic.

It is shown that the equivalent Nyquist characteristic X_e(w ) for a given characteristic X(w ) is given by the equation

The equivalent Nyquist characteristic is constructed by slicing the original X(w ) into segments of width 2p /T and superimposing all the segments on the interval [-p/T, p/T].

For no interference, that means to have x_n=0 for n¹ 0, the equivalent Nyquist characteristic (Figure 2.12) is

The characteristic (2.7) is the only meeting the conditions for no intersymbol interference because, in accord with the sampling theorem, being bandlimited at the Nyquist frequency (minimum bandwidth) is uniquely determined by the samples { x_n}.

Once, because it corresponds to no symbol interference, and twice, because it is not physically realizable. x(t), the response of the transmission system to an input pulse g(t), begins to exist before applying g(t). Practically it is necessary to approximate, as well as possible, the rectangular X(w ) in realizing the transmission system, if we want to use the minimum bandwidth corresponding, theoretically, to no intersymbol interference.

However, in nearly all practical cases of interest the actual bandwidth available is larger than the minimum-required Nyquist bandwidth for the desired symbol rate 1/T, but it does not exceed twice this bandwidth. If this restriction is made, i.e., if

constructing X_e (w ) is considerably simplified. This is illustrated in Figure 2.13 which shows a frequency characteristic X(w ) assumed to be a real function of w .

The equivalent Nyquist bandwidth characteristic is obtained by superimposing the three bands labeled X-₁, X₀, X₁. X₁ has no components for positive frequencies when superimposed on X₀. Placing X-₁ on X₀ is equivalent to "folding" X(w ) back onto itself about the Nyquist frequency p/T. To have no intersymbol interference, the equivalent Nyquist bandwidth characteristic which can be obtained in this way must be rectangular. For this, the characteristic X(w ), when it is a real one, must have an odd symmetry about w=w_N.

Obviously if the bandwidth is larger than the Nyquist band the criterion of removal of intersymbol interference does not uniquely specify the pulse spectrum X(w ). The ambiguity involved in this case must be resolved using other considerations. Such considerations are related to the rate of decay with time of the pulse x(t) and the possibility to approximate better in a practical implementation the ideal, nonrealizable, characteristic X(w ).

The error in approximation of X(w ) with a real system and the jitter (fluctuation) of sampling instants around the ideal instants will result in nonzero values of the x(t) at the actual pulse sampling times. Generally speaking, the faster x(t) decays with time, the less is the effect of the unavoidable perturbations. If, for example, X(w ) is rectangular, then x(t) decreases as 1/t for large t. One class of Nyquist characteristics which has been extensively used and studied is the so called raised cosine characteristic. A raised cosine characteristic consists of a flat amplitude portion and a roll-off portion that has a sinusoidal form (Figure 2.14):

The response x(t) is given by:

ais a parameter, called roll-off factor, which indicates the ratio between the supplementary bandwidth used in excess of the minimum Nyquist bandwidth and the Nyquist bandwidth. It is noticed that x(t) decreases asymptotically as 1/t³.

The overall pulse spectrum X(w ) being chosen to meet the Nyquist condition assuring no intersymbol interference under perfect timing conditions, there still remains an arbitrary choice of transmitting or receiving filters as X(w )=G(w )G_T(w )C(w )G_R(w ) (see Figure 2.8). Supposing a perfect channel (without amplitude and phase distortions, for example C(w )=1) with a flat spectral density noise, it is shown that maximum noise - immunity is obtained if the overall characteristic X(w ) is equally splitted between the transmitter and receiver: