The image you've uploaded represents a discrete signal $x[n]$. The values are shown at specific positions along the horizontal axis, with the vertical axis indicating the amplitude of the signal.

Here are the three modifications requested based on the original $x[n]$:

1. $x[n-3]$:

This represents a right shift of the original signal by 3 units. Each point on the signal $x[n]$ will be shifted to the right by 3 positions.

• Every value of $x[n]$ is moved 3 units to the right.
• Example: The value originally at $n = -2$ will move to $n = 1$, the value at $n = 0$ will move to $n = 3$, etc.

2. $x[-n+3]$:

This involves time reversal followed by a shift to the right by 3 units.

• First, reverse the signal (flip around $n = 0$), so the values that were on the left side of the origin will move to the right side, and vice versa.
• After reversing, shift the whole signal 3 units to the right.
• Example: The value at $n = -2$ becomes the value at $n = 2$, and then shift it right to $n = 5$.

3. $x[3n]$:

This represents compression by a factor of 3. The signal will be compressed along the $n$-axis, meaning that only the points where $n$ is a multiple of 3 will retain their values, and all other points will have a value of 0.

• The signal will be "sped up" or "compressed."
• Example: Values originally at $n = 0$, $n = 3$, $n = -3$, etc., will remain, while intermediate points like $n = 1, 2, -1$ will become zero.

Visual Representation:

1. $x[n-3]$:

Shift the entire signal to the right by 3 positions.

2. $x[-n+3]$:

Flip the signal horizontally, then shift it right by 3.

3. $x[3n]$:

Only retain points at positions $n = 0, \pm3, \pm6$, with others zeroed.

Would you like a graphical visualization or further details on how to sketch these signals?

Related Questions:

1. What is the impact of shifting signals on their Fourier transform?
2. How would the signal change if $x[n+3]$ (left shift) is applied instead of $x[n-3]$?
3. How does time reversal ($x[-n]$) affect the symmetry of a signal?
4. What happens if we apply a scaling factor other than 3 in $x[kn]$?
5. How does the compression of a signal affect its frequency spectrum?

Tip:

When modifying signals, always pay attention to whether you're shifting left or right, compressing, or reversing—it significantly alters the time-domain characteristics.