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I want to know that if I have any signal given how will I draw it according to the following equation

$$x(t-1)-1/2$$

what will be the role of $1/2$ when drawing the new signal?

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    $\begingroup$ you shift your signal by 1 second to the right and shift it down by 1/2 $\endgroup$ – Ben Nov 1 '18 at 13:59
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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 $t$.

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  • $\begingroup$ Neat, neat, neat $\endgroup$ – Laurent Duval Nov 1 '18 at 14:31
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$x(t+\tau)$ 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

% 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')
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