**Time shifting operations :**

The different time-shifting operations are as follow :

- Time delay
- Folding
- Time advance
- Folding
- Folding and advance
- Folding and delay

**1. Time delay :**

In the case of a discrete time signal, the given sequence can be delayed by a few samples. We know that the discrete time signal is denoted by x (n).

Suppose we want to delay this sequence by "k" sample. It will be denoted by x (n-k).

x (n) = Original sequence

x (n-k) = Original sequence delayed by k samples.

Here k is an integer

x(n) = {1, 2, 3, 4, 5 }, k=2

↑

x(n-2) = { 0, 0,1, 2, 3, 4, 5 }

↑

**2. Time advance :**

Time advance operation is opposite to the time delay operation. Consider the same sequence is shown given below :

x(n) = {1, 2, 3, 4, 5 }

↑

x(n+2) = { 1, 2, 3, 4, 5 }

↑

**3. Folding :**

**Folding is also called as reflection. Thus if x (n) represents input signal then x (-n) represent folded input signal.**

x (n) = { 1, 2, 3, 4, 5 }

x (-n) = { 5, 4, 3, 2, 1 }

↑

**4. Folding and delay :**

- First fold the sequence x(n); that means obtain x (-n)
- Then delay the folded sequence by k sample

delay

x (n) → x (n-k)

delay

x (-n) → x [- (n-k) ] = x (-n+k)

x (n) = { 1, 2, 3, 4, 5 }

↑

x (-n) = { 5, 4, 3, 2, 1 }

↑

x (-n+2) = { 5, 4, 3, 2, 1 }

↑

**5. Folding and advance :**

advance

x (n) → x (n+k)

advance

x (-n) → x [- (n+k) ] = x (-n-k)

x (n) = { 1, 2, 3, 4, 5 }

↑

x (-n) = { 5, 4, 3, 2, 1 }

↑

x(-n-2) = { 5, 4, 3, 2, 1, 0, 0 }

↑

**Time scaling operation :**

- Downscaling
- upscaling

**1. Down scaling :**

consider the same sequence x (n) = { 1, 2, 3, 4, 5 }

↑

y (n) = x (2n)

Now from given sequence x (n) we can write :

x(0) = 1

x(1) = 2

x(2) = 3

x(3) = 4

x(4) = 5

This gives the value of x (n) for different value of n;

y(n) = x (2n)

y(0) = x (0) = 1

y(1) = x (2) = 3

y(2) = x (4) = 5

y(3) = x (6) = 0

y(n) = x (2n) = {1, 3, 5, 0.....}

↑

**2. Up scaling or expansion :**

Consider same input sequence x (n) = { 1, 2, 3, 4, 5 } is applied to certain device which produces output y(n) = x(n/2). ↑

Thus in this case :

y(n) = x(n/2)

y(0) = x(0/2) = x(0) =1

y(1) = x(1/2) → No sample

y(2) = x(2/2) = x (1) = 2

y(3) = x(3/2)= x(1.5) → No sample

y(4) = x(4/2) = x (2) = 3

y(5) = x(5/2) = x (2.5) → No sample

y(6) = x(6/2) = x (3) = 4

y(7) = x(7/2) = x (3.5) → No sample

y(8) = x(8/2) = x (4) = 5

y (n) = x (n/2) = { 1, 0, 2, 0, 3, 0, 4, 0, 5 }

↑

**Amplitude scaling operation :**

- Up-scaling
- Down-scaling
- Addition
- Multiplication

**1. Up-scaling :**

x (n) = { 1, 2, 3, 4, 5 }

↑

x(0) = 1

x(1) = 2

x(2) = 3

x(3) = 4

x(4) = 5

y (n) 2 x(n) = { 2, 4, 6, 8, 10 }

↑

**2. Down-scaling :**

x (n) = { 1, 2, 3, 4, 5 }

↑

y (n) = x (n) / 2 = { 0.5, 1, 1.5, 2, 2.5 }

↑

**3. Addition :**

x

_{1}(n) = { 1, 1, 0, 1, 1 }
↑

x

↑_{2}(n) = { 2, 2, 0, 2, 2 }y (n) = x

_{1}(n) + x

_{2}(n)

y (n) = { 3, 3, 0, 3, 3 }

↑

**4. Multiplication :**

x

_{1}(n) = { 1, 1, 0, 1, 1 }
↑

x

↑_{2}(n) = { 2, 2, 0, 2, 2 }y (n) = x

_{1}(n) * x

_{2}(n)

y (n) = { 2, 2, 0, 2, 2 }

↑