Question: I'm studying for an exam and have some trouble with an exercise.
So I have a continuous-time signal that is

$$s(t)=\sin(2\pi f\_{s1} t) + \sin(2\pi f\_{s2}t)$$
with $f\_{s1} = 1.4\text{ kHz}$ and $f\_{s2} = 1.45\text{ kHz}$.

After sampling the time continuous signal I'll have to get the DFT of the discrete signal. The task is to chose the sampling frequency and the base interval of the discrete signal so that there will be no leakage-effect after calculating the DFT.

I also know that $\frac{N}{\text{period}}$ has to be an integer.

How would you approach something like this? I don't expect to be spoonfed the answer, hints would be nice though.

Answer: 
A signal $x(t)$ is a function that maps a time instant $t$ to a corresponding value. Thus, the manipulations that you described in your question apply as they would with any other function:

* $x(t-1)$ will effect a right shift along the $t$ axis by $1$ unit.
* $-\frac{1}{2}$ will effect a downward shift along the $x(t)$ axis by $\frac{1}{2}$ units.

That is, the role of $-\frac{1}{2}$ is to just subtract that value from the signal at all values of $t$.
$x(t+\tau)$ moves the signal to the left if $\tau$ is positive, because $t+\tau\ge t$ comes "earlier". And to the right if $\tau$ is negative. $x(t)+y\_0$ moves to the top if $y\_0$ is positive, and downward if $y\_0$ is negative. Combining both, with a visual version using Matlab:

[![shifted signal](https://i.stack.imgur.com/ROu1m.png)](https://i.stack.imgur.com/ROu1m.png)

```
% StackExchange, DSP (Signal Processing) 53002
% https://dsp.stackexchange.com/questions/53002/how-will-i-draw-the-following-signal
% Laurent Duval
% Creation: 2018/11/01
% Update: 2018/11/02

clear all;close all
signalOriginal = @(t) ((abs(t) < 4) .*sinc(t));
timeShift = -1;
valueShift = -1/2;

nSample = 1024;
timeStart = -2;
timeStop = 2;

timeOriginal = linspace(timeStart,timeStop,nSample)';
timeShifted = linspace(timeStart+timeShift,timeStop+timeShift,nSample)';

dataOriginal = signalOriginal(timeOriginal);
dataShifted = signalOriginal(timeShifted)+valueShift;

figure(1);clf;hold on;
plot(timeOriginal,dataOriginal,'b')
% plot(-[0 timeShift],[0 valueShift]+max(dataOriginal),'k-')
quiver(0,1,-timeShift,0,'k:')
quiver(0,1,0,valueShift,'k--')
quiver(0,1,-timeShift,valueShift,'k-')
plot(timeOriginal,dataShifted,'r')
grid on
hold off
legend('Original','Time shift','Value shift','T-V shift','Shifted')

```