10.6 Discrete time processing of continuous time signals  (Page 3/3)

Anti-aliasing filter

In reality, we cannot typically guarantee that the input signal will have a specific bandlimit, and sufficiently high sampling rates cannot necessarily be produced. Since it is imperative that the higher frequency components not be allowed to masquerade as lower frequency components through aliasing, anti-aliasing filters with cutoff frequency less than or equal to ${\omega }_s/2$ must be used before the signal is fed into the ADC. The block diagram in [link] reflects this addition.

As described in the previous section, an ideal lowpass filter removing all energy at frequencies above ${\omega }_s/2$ would be optimal. Of course, this is not achievable, so approximations of the ideal lowpass filter with low gain above ${\omega }_s/2$ must be accepted. This means that some aliasing is inevitable, but it can be reduced to a mostly insignificant level.

Signal quantization

In our preceding discussion of discrete time processing of continuous time signals, we had assumed an ideal case in which the ADC performs sampling exactly. However, while an ADC does convert a continuous time signal to a discrete time signal, it also must convert analog values to digital values for use in a digital logic device, a phenomenon called quantization. The ADC subsystem of the block diagram in [link] reflects this addition.

The data obtained by the ADC must be stored in finitely many bits inside a digital logic device. Thus, there are only finitely many values that a digital sample can take, specifically $2^N$ where $N$ is the number of bits, while there are uncountably many values an analog sample can take. Hence something must be lost in the quantization process. The result is that quantization limits both the range and precision of the output of the ADC. Both are finite, and improving one at constant number of bits requires sacrificing quality in the other.

Filter implementability

In real world circumstances, if the input signal is a function of time, the future values of the signal cannot be used to calculate the output. Thus, the digital filter $H_2$ and the overall system $H_1$ must be causal. The filter annotation in [link] reflects this addition. If the desired system is not causal but has impulse response equal to zero before some time $t_0$ , a delay can be introduced to make it causal. However, if this delay is excessive or the impulse response has infinite length, a windowing scheme becomes necessary in order to practically solve the problem. Multiplying by a window to decrease the length of the impulse response can reduce the necessary delay and decrease computational requirements.

Take, for instance the case of the ideal lowpass filter. It is acausal and infinite in length in both directions. Thus, we must satisfy ourselves with an approximation. One might suggest that these approximations could be achieved by truncating the sinc impulse response of the lowpass filter at one of its zeros, effectively windowing it with a rectangular pulse. However, doing so would produce poor results in the frequency domain as the resulting convolution would significantly spread the signal energy. Other windowing functions, of which there are many, spread the signal less in the frequency domain and are thus much more useful for producing these approximations.

Anti-imaging filter

In our preceding discussion of discrete time processing of continuous time signals, we had assumed an ideal case in which the DAC performs perfect reconstruction. However, when considering practical matters, it is important to remember that the sinc function, which is used for Whittaker-Shannon interpolation, is infinite in length and acausal. Hence, it would be impossible for an DAC to implement perfect reconstruction.

Instead, the DAC implements a causal zero order hold or other simple reconstruction scheme with respect to the sampling rate ${\omega }_s$ used by the ADC. However, doing so will result in a function that is not bandlimited to $(-{\omega }_s/2,{\omega }_s/2)$ . Therefore, an additional lowpass filter, called an anti-imaging filter, must be applied to the output. The process illustrated in [link] reflects these additions. The anti-imaging filter attempts to bandlimit the signal to $(-{\omega }_s/2,{\omega }_s/2)$ , so an ideal lowpass filter would be optimal. However, as has already been stated, this is not possible. Therefore, approximations of the ideal lowpass filter with low gain above ${\omega }_s/2$ must be accepted. The anti-imaging filter typically has the same characteristics as the anti-aliasing filter.

Discrete time processing of continuous time signals summary

As has been show, the sampling and reconstruction can be used to implement continuous time systems using discrete time systems, which is very powerful due to the versatility, flexibility, and speed of digital computers. However, there are a large number of practical considerations that must be taken into account when attempting to accomplish this, including quantization noise and anti-aliasing in the analog to digital converter, filter implementability in the discrete time filter, and reconstruction windowing and associated issues in the digital to analog converter. Many modern technologies address these issues and make use of this process.