• A low pass signal contains frequencies from 1 Hz to some higher value.

• Statement of the sampling theorem

1) A band limited signal of finite energy , which has no frequency components higher than W hertz , is completely described by specifying the values of the signal at instants of time separated by $\frac{1}{2W}$ seconds and

2) A band limited signal of finite energy, which has no frequency components higher than W hertz , may be completely recovered from the knowledge of its samples taken at the rate of 2W samples per second.

The first part of above statement tells about sampling of the signal and second part tells about reconstruction of the signal. Above statement can be combined and stated alternately as follows :

Acontinuous time signal can be completely represented ints samples and recovered back ifthe sampling frequency is twice of the highest frequency content of the signal i.e.,

$f_s≥ 2W$ Here $f_s$ is the sampling frequency

And W is the higher frequency content

Proof of sampling theorem

There are two parts :

I)Representation of x(t) in terms of its samples

II)Reconstruction of x(t) from its samples

PART I: Representation of x(t) in its samples $x(nT_s)$

Step 1 : Define$ x_δ (t)$

Step 2 : Fourier transform of $x_δ (t)$ i.e. $X_δ (f)$

Step 3: Relation between X(f) and $X_δ (f)$

Step 4 : Relation between x(t) and $x(nT_s)$

Step 1 : Define$ x_δ (t)$

The sampled signal $x_δ $(t) is given as ,

Here, observe that $x_δ (t)$ is the product of x(t) and impulse train δ(t) as shown in figure.

In the above equation δ(t-$nT_s$) indicates the samples placed at ±$T_s$,±$2_s$,±$3T_s$… and so on

Step 2 : Fourier transform of $x_δ$ (t)i.e. $ X_δ (f)$

Taking FT of equation (1)

We know that FT of product in time domain becomes convolution in frequency domain i.e.,

Conclusions:

i)The RHS of above equation shows that X(f) is placed $at ±f_s,±2f_s,±3f_s,..$

ii)This means X(f) is periodic in $f_s$.

iii) If sampling frequency is$ f_s$=2W , then the spectrums X(f) just touch each other .

Step 3:Relation between X(f) and $X_δ$ (f)

Important assumption

Let us assume that $f_s$=2W ,then as per above diagram .

In above equation ‘f' is frequency of CT signal .And $\frac{f}{f_s}$ = Frequency of DT signal in equation (4) .Since x(n)=x($nT_s$ ),i.e. samples of x(t),thenwehave,

Since $\frac{1}{f_s} =T_s$

Putting above expression in equation (3),

Conclusions:

1) Herex(t) is represented completely interms of x$(nT_s)$.

2) Above equation holds for$ f_s$=2W.This means if the samples are taken at the rate of 2W or higher, x(t) is completely represented by its samples.

3) First part of the sampling theorem is proved by above two conclusions.

II) Reconstruction of x(t)from its samples

Step 1 : Take inverse Fourier transform of X(f) which is in terms of $X_δ$ (f)

Step 2 : Show that x(t) is obtained back with the help of interpolation function.

Step 1 :Take inverse Fourier transform of equation (5) becomes ,

Interchanging the order of summation and integration,

Simplifying above equation,

Conclusions:

The samples x$(nT_s $)are weighted by sinc functions.

The sinc function is the interpolating function .fig 4.1.3 shows, how x(t)is interpolated.

Step 3:

Reconstuction of x(t) by low pass filter

When the interpolated signal of equation (6) is passed through the low pass filter of bandwidth -W≤f≤W , then the reconstructed waveform shown in fig.4.1.3(b) is obtained.The individual sinc functions are interpolated to get smooth x(t).

When high frequency interferes with low frequency and appears as low frequency , then the phenomenon is called aliasing.

Effects of aliasing:

i)since high and low frequencies interfere with each other , distortion is generated.

ii)The data is lost and it cannot be recovered.

Different ways to avoid aliasing :

Aliasing can be avoided by two methods

i)sampling rate $f_s≥2W$

ii)Strictly bandlimit the signal to ‘W’