$x(t)$ and $x(-t)$ are flipped (left to right) signals of each other. but $x(2t-3)$ and $x(-2t+3)$ do not share that kind of relation as I thought incorrectly. Can some one comment on this? Is there a name for the relationship of this kind of signals? Is it important enough to study this relation and give a name?

  • $\begingroup$ this is a common conceptual problem when people first learn about convolution. at least it was for me. $\endgroup$ Jun 4 '18 at 16:48

$x(t)$ and $x(-t)$ are flipped left-to-right about the point $t=0$. that point of reflection at $t=0$ is determined by asking: "When is $t$ and $-t$ equal to the same value?"

$$ t = -t \qquad \qquad \implies t=0 $$

so it turns out that $x(2t-3)$ and $x(-2t+3)$ are also flipped left-to-right, but the point of reflection is not $t=0$. So then, what value of $t$ is the point of reflection and how do we determine that value? Like above, it is when the two expression of argument to the common function are equal to each other:

$$ 2t-3 = -2t+3 $$

solve for $t$ and you'll find your point of reflection.

  • $\begingroup$ I realize x(2t-3) and x(-2t-3) are flips of each other. So x(-2t+3) is flip and shift of x(2t-3). $\endgroup$ Jun 5 '18 at 4:50

What you are describing refers to symmetry.

$x(t)$ and $x(-t)$ are flipped (left to right) signals of each other...

$x(t)$ seems to be referring to a function with infinite support extending both towards positive and negative infinity. if $t$ starts at $0$ and increasing, then putting a minus sign indeed reverses its direction. However, unless you define that a signal in $x$ is supposed to have infinite support (?), looking for values at $t=-6.0$ is an invalid operation.

You could of course check the value of something like a $\cos(-2t+3)$ because we know that the $\cos$ function extends to infinity.

A signal would be an indexed sequence of measurements in discrete time. For example, $x = \left[ 0,1,2,3,0,1,2,3,0,1,2,3,0\right]$ sampled at some $Fs$ and having a $\Delta t = \frac{1}{Fs}$. Therefore, index $n \cdot \Delta t$ with $n>0$ now refers to one of the numbers within the range of the array. We usually drop the multiplication with $\Delta t$ as implied and use the notation $x[n]$ for discrete-signals.

With this in mind:

$x[n]$ and $x[N_x-n-1]$, where $N_x = |x|$ (i.e. the length of sequence $x$) and for $n \in [0 .. N_x-1]$ would have even symmetry.

The same (even symmetry) should hold for $2n-3$ because the numbers generated for $n=0$ and increasing by $1$ would only differ in their sign. But, assuming that this is now the index to an array, its "reversal" would have to be adjusted for the length of the array.

Hope this helps


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