# Difference between two Sine wave equations

I'm confused as what is the difference between these two sine wave equations

x(t) = Sin(2*pi*F*t)


and

x(t) = Sin(2*pi*F*nTs)


x(t) = Sin(2*pi*F*t) is supposed to be used with analog signal, here t is supposed to me the samples when speaking of matlab.

A = 4;       %Amplitude
Freq = 100;
Time = 1/Freq; %0.01
Fs = 1000;      %Sampling Frequency
Ts = 1/Fs;   %Sampling Rate
t = 0:Ts:(Time);
x=A*sin(2*pi*Freq*t);


The output, Similarly if I use the stem command than a plot command I will get the sampled values instead. If I'm getting the sampled result from the first equation than what is the purpose to replace t by nTs in the equation and using the second equation ?

I couldn't be able to demonstrate the difference and visualize it in Matlab. Kindly guide me.

## 1 Answer

The first equation:

x(t) = Sin(2*pi*F*t)


is for a continuous ("analogue") signal, where t may take any real value.

The second equation:

x(t) = Sin(2*pi*F*n*Ts)


(I assume that this is the correct version - it looks like you have a typo in the question ?)

is for a discrete ("sampled" or "digital") signal, where n is the sample number (integer) and Ts is the sample interval (inverse of sample rate). It has values only at the discrete sampling points where t = n * Ts.

In your MATLAB code you are effectively sampling a continuous function, so although you start off with the first (continuous) equation you end up with the second equation because you are only evaluating (sampling) the continuous function at a set of discrete points.

Note also that as Ts -> 0 the discrete version tends towards the continuous version.