Polar modulation
Polar modulation is analogous to quadrature modulation. Quadrature modulation makes use of the cartesian coordinates x and y, while polar modulation makes use of the polar coordinates, amplitude (r), and phase (∅). To create a polar signal, the phase transfer of the amplifier should be at least over a 17dB range, as the phase transitions from one to another there will be amplitude perturbation.
Sampling.
We have widely discussed the sampling process, including a derivation of the sampling theorem and related issues of aliasing and reconstructing the message signal from its sampled version in our previous blogs, and now we will be restating the sampling theorem in the context of PAM in two equivalent parts as follows:
- A band-limited signal of finite energy that has no radian frequency components higher than wm is uniquely determined by the values of the signal at instants of time separated by pi/wm seconds.
- A band-limited signal of finite energy that has no radian frequency components higher than wm may be completely recovered from a knowledge of its samples taken at the rate of wm/ pi per second.
Typically, the spectrum of a message signal is not strictly banded limited, contrary to what is required by the sampling theorem. Rather, it approaches zero asymptotically as the frequency approaches infinity, which gives rise to aliasing and therefore distorts the signal. As we know that aliasing consists of a high-frequency component in the spectrum of the message signal apparently taking on the identity of a lower frequency in the spectrum of a sampled version of the message signal. To combat the effects of aliasing in practice, we use two corrective measures:
- Prior to sampling, a low-pass anti-aliasing filter is used to attenuate those high-frequency components of the signal which lie outside the band of interest.
- The filtered signal is sampled at a rate slightly higher than the Nyquist rate.
Mathematical description of PAM
The carrier wave used in PAM consists of a sequence of short pulses of fixed duration in terms of which PAM is formally defined as follows: PAM is a form of pulse modulation in which the amplitude of the pulsed carrier is varied in accordance with instantaneous sample values of the message signal; the duration of the pulsed carrier is maintained constant throughout. The fundamental frequency of the carrier wave (i.e. the pulse repetition frequency) is the same as the sampling rate. For a mathematical representation of the PAM signal s(t) for a message signal m(t) we may write:
s(t) = Σ m[n] h(t-nT).
where T is the sampling period, m[n] is the value of the message signal m(t) at time t= nT and h(t) is a rectangular pulse of unit amplitude and duration T.
In physical terms, the equation represents a “sample and hold” operation analogous to the “zero-order hold-based” reconstruction.
Suppose that m(t) is strictly band limited and that the sampling rate 1/ T, is, greater than the Nyquist rate. Then passing s(t) through a reconstruction filter chosen as an ideal low-pass filter with cutoff frequency and gain T, we find that the resulting filter is equal to M(jw) H(jw). This result is equivalent to that which would be obtained by passing the original message signal m(t) through a low-pass filter with frequency response H(jw).
By using PAM to represent a continuous-time message signal, we introduce amplitude distortion as well as a delay of T/2. Both of these effects are present as well in the sample-and-hold reconstruction scheme. A similar form of amplitude distortion is caused by the finite size of the scanning aperture in television and facsimile. Accordingly, the frequency distortion caused by the use of flat-topped samples in the generation of a PAM wave is referred to as the aperture effect.
DEMODULATION OF PAM SIGNAL
The system consists of two components connected in a cascade. The first component is a low-pass filter with a cutoff frequency that equals the highest frequency component omega – of the message signal. The second component is an equalizer that corrects for the aperture effect due to flat-topped sampling in the sample-and-hold circuit. The equalizer has the effect of decreasing the in-band loss of the interpolation filter as the frequency increases in such a manner as to compensate for the aperture effect.
Reference