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# What is the Aliasing effect?

Whenever a signal is sampled below Nyquist rate, the signal is called under sampled and the Aliasing effect hampers the results. There are anti aliasing filters used to resolve this issue. Below we discuss the concept in detail.

When the sampling frequency is greater than or equal to twice the highest frequency component of the message signal, then any continuous time signal can be recovered. This is written as

fs ≥ 2fm————————-(1)

Let x(t) be a continuous time signal. For signal x(t) the spectrum is 0 for |ω|> ωm. As the signal is band limited to a frequency of fm Hz.

When an impulse signal δ(t) is multiplied with x(t), a sampled signal is obtained having time period Ts. The figure shows signal y(t) which is the output signal. It is obtained as a result of multiplication of signals x(t) and δ(t).

As a result of sampling the sampled output also has the same time period as of impulse. The equations below shows

When we put the values in equation 2 we get

Now we take Fourier Transform of both sides

In order to recover the input signal we need to recover the input signal spectrum  X(ω) from the output signal Y(ω). This can only happen when there is no overlapping in Y(ω). Below figure explains the conditions for different sampling conditions.

The overlapped region in case of under sampling represents aliasing effect, which can be removed by

• Considering fs >2fm
• By using anti-aliasing filters.

When fs =2fm it is called Nyquist criteria

## Nyquist Rate

It is the minimum sampling rate at which signal can be converted into samples and recovered back without distortion.

• Nyquist rate fN = 2fm hz
• Nyquist interval = 1/fN = 1/2fm seconds.

## Aperture effect

• The amplitude of the flat top signal must be constant, but sometimes it is not constant due to the high frequency roll off of the sampling signal.
• As a result of which there is attenuation in the high frequency part of the message spectrum.
• Thus, the sampled signal in the flat top sampling consists of attenuated high frequency components and this effect is known as Aperture effect.
• Aperture effect can be improved by selecting the value of pulse width τ to be very small and by using an equalizer circuit.

If the waveform is under sampled (when. fs < 2B) then the signal at the output of the filter will be different from the original signal spectrum This results in aliasing

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