Question

A digital signal has 16 possible levels. How many bits are necessary to represent each level?

Answer 1

Why is the Bandwidth of digital signal infinite?

The Bandwidth of a digital signal is very very large, tending to infinite.

To understand this , let’s consider a digital signal of finite duration and a certain amplitude.

By looking at this, we may have an initial impression that this signal which is of some fixed frequency cannot have an infinite bandwidth. But, remember every signal is the combination(summation) of many sinusoidal frequencies. This is what Fourier Series says.

So, now let’s use the Fourier series representation of the square wave.

It approximately looks like this (considering the base frequency and some harmonics)

Observe, the summation of the sinusoidal frequencies at each point gives the square wave (approximately).

Now, to make it more precise, consider more harmonics that occur in the Fourier series expression.

Even the summation of sine amplitudes at every point in the above diagram cannot exactly represent the square wave. The presence of harmonics goes on up to infinite number of sine waves with different values of frequencies.

So, the difference of the frequencies of the base sine wave and the the sine wave of lowest frequency will give the bandwidth , which would be very very high.The Fourier components present in the square waves can be easily visualized by this figure.

This condition arise because of the discontinuity of the square wave, because of the straight line at the boundary of the square wave and its sharp edges. So, how much ever we go on considering the number of harmonics, we cannot obtain a perfect straight edge. It will always have smooth edges.

The digital signal practically looks like this :

Hope this helps :)

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In signal processing, there is a time-frequency duality.

Band-limited signals have infinite time duration and time-limited signals have infinite bandwidth.

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first of all understand what a digital signal is. the first step of and digitisation is antialiasing and sampling. why antialiasing? such that you can correctly sample at exact of higher than nyquist rate. am i correct? let’s now move on.

your bandwidth issues come at sampling. what exactly is sampling? it is nothing but multiplication with an impulse train. an infinitely long impulse train. now what will be the fourier transform of an impulse train? again an impulse train at intervals of Fs. am i correct?

let’s take first this ideal case where infinitely long impulse train you are multiplying with your time domain signal. what is the equivalent frequency domain operation? it is nothing but convolution of both the spectrum. convolving any finite bandwidth signal with an impulse train of interval Fs (i. e. at every integer multiple of Fs, you have an impulse) results in an infinite bandwidth signal. read these few lines once more and try to visualize.

Answer 2

4 bits are necessary to represent each level.

Since it is given that the digital signal has 16 possible levels. So the necessary bits can be calculated by the log function with base 2. It is the formula for finding the level of signals.

Now the log with base two of sixteen is,

                                           $=log_2(16)\\=log_2(2*2*2*2)\\= 4$

Hence the necessary bits to represent each level is 4 bits.