Ideal Low Pass Filter - Electrical Engineering (EE) PDF Download

How can we tackle the problem of Low pass Filter not being ideal ?

  =Magnitude Response of the filter

=Phase Response
Normally we want ,  to be zero .(This is what we want IDEALLY) 

Next best thing that we can do is we can have some linear phase variation, i.e. constant time delay for all frequencies.
                             

Unfortunaltely analog filters can NEVER give linear phase response.

We can design analog filters as near to an ideal filter in terms of magnitude response, but can not really make ideal filter.

How can we solve this problem?

What we have to do is to get a Maximally Flat Sampling, i.e. fs>>2fm

This is what we can do using Hold Filters, which is referred to as Zero - Order - Hold Sampling. It is a staircase approximation of the analog signal.

How it works?

In practice, analog signals are sampled using zero - order - hold (ZOH) devices that hold a sample value constant until the next sample is acquired. This is also called as flat - top sampling. This operation is equivalent to ideal sampling followed by a system whose impulse response is a pulse of unit height and duration T s ( to stretch the incoming pulses ). This is illustrated in Figure below :



 

Reconstruction of signal in Zero Order Hold Filter

The analog Signal (continuous - time signal) is multiplied with a periodic impulse train, referred to as Sampling Function. A sampled signal is then obtained as shown in figure
below.

The ideally sampled signal xp(t) is the product of the impulse train p(t) and the analog signal xc(t) and is written as

The ZOH Sampled Signal x_ZOH(t) can be regarded as the convolution of h_o(t) and a sampled signal x_p(t)

Distortion in Zero-order-hold sampling :

 The transfer function H(f) of the zero - order - hold circuit is the Sinc function

Since the spectrum of the ideally sampled signal is 
 The spectrum of the zero- order - hold sampled signal xZOH(t) is given by the product

This spectrum is illustrated in Figure shown below :

Figure : Spectrum of a zero - order - hold sampled signal The term sinc( f / f_s ) attenuates the spectral images X( f - k f_s ) and causes their distortion.

There are two types of distortion :-

 a) Aliased Component Distortion : Aliased Component distortion can be corrected, if required by cascading another better lowpass filter.
 b) Baseband Spectrum Distortion (Sinc Distortion) : Baseband Spectrum Distortion is corrected by an Equalizer. An Equalizer is an LSI system with Fourier Transformable impulse response which acts like an inverse 1 / H ( f ) to another LSI system, at least in a certain range of frequencies. Equalizer is also used to correct channel imperfections in a communication system.

The higher the sampling rate f_s, the less is the distortion in the spectral image X( f ) centered at origin.

 An ideal lowpass filter with unity gain over -0.5 fs≤ f ≤ 0.5 fs recovers the distorted signal.

To recover X( f ) with no amplitude distortion, we must use a compensating filter that negates the effects of the Sinc distortion by profiling a concave shaped magnitude spectrum corresponding to the reciprocal of the Sinc function over the principal period | f | ≤ 0.5 fs
 

Figure : Spectrum of a filter that compensates for Sinc distortion The magnitude spectrum of the compensating filter is given by

Reconstruction of signal in Zero Order Hold Filter The analog Signal (continuous - time signal) is multiplied with a periodic impulse train, referred to as Sampling Function. A sampled signal is then obatained as shown in figure below.
 

The ideally sampled signal xp(t) is the product of the impulse train p(t) and the analog signal xc(t) and may be written as

The ZOH Sampled Signal xZOH (t) can be regarded as the convolution of ho(t) and a sampled signal xp(t)

Conclusion:

In this lecture you have learnt:

• analog filters can NEVER give linear phase response .
• Hence, we can design analog filters as near to an ideal filter in terms of magnitude response, but can not really make ideal filter .
• Hold Filters can be used to get approximation by a Maximally Flat Sampling i.e fs >> 2fm.

There are 2 types of distortion: Baseband Spectrum Distortion (Sinc Distortion) & Aliased Component Distortion.

The ZOH Sampled Signal xZOH(t) can be regarded as the convolution of h_o(t) and a sampled signal x_p(t).