Continuous-time Signals ELEC 309 Prof. Siripong Potisuk

Signal Transformations Operations Performed on the Independent and Dependent Variables 1) Reflection or Time Reversal or Folding 2) Time Shifting 3) Time Scaling 4) Amplitude Scaling 5) Amplitude Shifting Note: The independent variable is assumed to be t representing time.

Time reversal or Folding or Reflection about t = 0 For CT signals, replace t by –t x(-t) is the reflected version of x(t) obtained from x(t) by a reflection about t =0

Time Shifting (Advance or Delay)

Time Scaling

Amplitude Scaling Multiply x(t) by A, where A is the scaling factor If A is negative, the original signal x(t) is also reflected about the horizontal axis.

Amplitude Shifting Add a constant A to x(t), where A is the amount of shift (upward or downward)

Signal Characteristics Deterministic vs. Random Finite-length vs. Infinite-length Right-sided/ Left-sided/ Two-sided Causal vs. Anti-causal Periodic vs. Aperiodic (Non-periodic) Real vs. Complex Conjugate-symmetric vs. Conjugate-antisymmetric Even vs. Odd

Even & Odd CT Signals A complex-valued signal x(t) is said to be A real-valued signal x(t) is said to be

Even-Odd Signal Decomposition A CT signal can be decomposed into its even and odd parts.

CT Periodic Signals A CT periodic signal x(t) is a function of time that satisfies the condition T is a positive constant T 0 is the smallest value of T called the fundamental period of x(t) f 0 = 1/T 0 called the fundamental frequency  0 = 2  f 0 called the angular frequency

A Sum of CT Periodic Signals

CT Aperiodic (nonperiodic) Signals Does not satisfy the condition Periodic extension accomplished by using summation and time-shifting operation

Signal Metrics Energy Power Magnitude Area

Energy An infinite-length signal with finite amplitude may or may not have finite energy. A finite-length signal with finite amplitude have finite energy.

Power A finite energy signal with zero average power is called an ENERGY signal. An infinite energy signal with finite average power is called a POWER signal.

Average Power Average over one period for periodic signal, e.g., Root-mean-square power:

Magnitude & Area