Signals and Systems
Lecture 1
Signals
A signal is a function representing a physical quantity
or variable, and typically it contains information about
the behavior or nature of the phenomenon.
Systems
A system is a mathematical model of a physical
process that relates the input (or excitation) signal to
the output (or response) signal.
Examples of Signals
Electroencephalogram (EEG) or Brain wave
Examples of Signals
Stock Market data as signal (time series)
Examples of Signals
Magnetic Resonance Image (MRI) data as 2-dimensional signal
Size of a Signal x(t) (1)
Measured by signal Energy Ex
Generalize for a complex valued signal
Energy must be finite which means:
Size of a Signal x(t) (2)
• If amplitude of x(t) does not 0 when t
need to measure power Px instead:
generalize for a complex valued signal to
∞,
Size of a Signal x(t) (3)
• Signal with finite energy (zero power)
Signal with finite power (infinite energy)
Useful Signal Operations
Time Shifting (1)
Signal might be delayed by
time T
∅ 𝑡 = 𝑥(𝑡 − 𝑇)
or advanced by time T:
∅ 𝑡 = 𝑥(𝑡 + 𝑇)
Time Scaling (2)
Signal may be compressed in time
(by a factor of 2):
∅ 𝑡 = 𝑥(2 𝑡)
or expanded in time
(by a factor of 2):
∅ 𝑡 = 𝑥(𝑡/2)
Same as recording played back at
twice and half the speed respectively
Time Reversal (3)
Signal may be reflected about the
vertical axis (i.e. time reversed):
∅ 𝑡 = 𝑥(−𝑡)
For example, the signal x(2t - 6) can
be obtained in two ways;
Delay x(t) by 6 to obtain x(t - 6),
and then time-compress this
signal by factor 2 (replace t with
2t) to obtain x (2t - 6).
• Alternately, time-compress x (t) by
factor 2 to obtain x (2t), then
delay this signal by 3 (replace t
with t - 3) to obtain x (2t - 6).
Amplitude scaling
• Amplitude scaling of a signal x(t) results in
amplification of x(t) if a >1, and attenuation if
a <1.
y(t) =ax(t)
Addition
• The addition of signals is given by
y(t) = x1(t) + x2(t)
Multiplication
• The multiplication of signals is given by
y(t) = x1(t).x2 (t)
Differentiation
• The differentiation of signals is given by
Integration
• The integration of a signal x(t) , is given by
Signals Classification (1)
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Signals may be classified into:
1. Continuous-time and discrete-time signals
2. Analogue and digital signals
3. Periodic and aperiodic signals
4. Energy and power signals
5. Deterministic and probabilistic signals
6. Causal and non-causal
7. Even and Odd signals
Signal Classification (1) – Continuous
vs Discrete
Continuous-time
Discrete-time
Signal Classification (2) – Analogue vs
Digital
Signal Classification (3) – Energy vs
Power
• x(t) (or x[n]) is said to be an energy signal (or
sequence) if and only if 0 < E < m, and so P = 0.
(Ex. Finite time signals)
• x(t) (or x[n]) is said to be a power signal (or
sequence) if and only if 0 < P < m, thus
implying that E = ∞.
(Ex. Periodic signals)
Signal Classification (4) – Periodic vs
Aperiodic
• A signal x(t) is said to be periodic if for some
positive constant To
• The smallest value of To that satisfies the periodicity condition
of this equation is the fundamental period of x(t).
Signal Classification (5) –
Deterministic vs Random
Deterministic
Random
Signal Classification (6) – Causal vs
Non-causal
A signal that does not start before t = 0 is a causal signal.
x(t) = 0 t < 0
A signal that starts before t = 0 is a noncausal signal.
A signal that is zero for all t ≥ 0 is called an anticausal signal.
Signal Classification (7) – Even vs Odd
• Even function
x(t) = x(-t)
cos(t)
odd function
x(t)=-x(-t)
sin(t)