1.2 Signal transformations involving linear transformations of the independent variable

Basic classes: Time Shift, Time Reversal, Time Scaling•Time Shift: Signals are identical in shape, but they are shifted relative to each other. Such as x[n] and x[n-n0], x(t) and x(t-t0).

• x[n-n0] is a delayed version of x[n] if n0 >0 .

• x(t-t0) is a advanced version of x(n) if t0 <0 .

• A time shift occurs only when the variable t or n are substituted for t-t0 or n-n0.

• x(at) and x(a(t-t0))= x(at-at0) are shifted relative to each other.

In applications: radar, sonar, communication and seismic signal processing.

Time Reversal: The signal x[-n] is obtained from the signal x[n] by a reflection about n=0. The signal x(-t) is obtained from the signal x(t) by a reflection about t=0.

• Not a causal operation

• A time reversal occurs only when the variable t or n are substituted for -t or -n.

• The signal x(-at+b) is obtained from the signal x(at+b) by a reflection about t=0.

• The signal x[-n+b]is obtained from the signal x[n+b] by a reflection about n=0.

Time Scaling: The signal x(at) (a>0) is obtained from the signal x(t) by linearly stretched if |a|<1,linearly compressed if |a|>1.

• Not a causal operation

• A time scaling occurs only when the variable t is substituted for at (a>0,a!=1).

• The signal x(at+b) (a>0) is obtained from the signal x(t+b) by linearly stretched if |a|<1,linearly compressed if |a|>1.

1.2 Signal transformations involving linear transformations of the independent variable

Basic classes: Time Shift, Time Reversal, Time Scaling

1.2.2 Periodic Signals• Definition

1.2.2 Periodic Signals• Fundamental Period

1.2.2 Periodic Signals

1.2.2 Periodic Signals

1.2.3 Even and Odd Signals

The Even-Odd Decomposition of an arbitrary Signal