Signals and Spectra
In the following we consider the representation of signals in the time and frequency domains,
and the relationship between these representations.
In the frequency domain we view the signal as consisting of sinusoidal components at various
frequencies.
The mathematical definition of the frequency domain representation, that is the spectrum, is
determined by the Fourier transform.
The spectrum for periodic signals is obtained using the Fourier series.
Sinusoidal signals
Sinusoidal signals are modelled as
v( t ) = A cos(ω 0 t + φ ) = A cos( 2 πf0 t + φ )
where A is the amplitude, ω 0 is the angular frequency ( f0 is the frequency) and φ is the phase.
This is a periodic signal whose period is T0 = 2 π / ω 0 = 1 / f0
One peak value of the signal is at t = − φ / ω 0
m
Spectra of sinusoidal signal
The line spectrum associates a certain frequency to a particular amplitude and phase.
The one sided line spectrum of a sinusoidal signal is:
The amplitude and phase spectrum have an impulse at f0 . The essential parameters of the signal can
be seen from the spectrum: frequency, amplitude, and phase.
g Kim
Linear combination of sinusoidal signals
w ( t ) = 7 − 10 cos( 40 πt − 60° ) + 4 sin 120 πt
This can be rewritten in the form:
w( t ) = 7 cos 2 π 0 t + 10 cos( 2 π 20 t + 120° ) + 4 cos( 2 π 60 t − 90° ),
which can be utilized to plot the one-sided line spectrum:
The complex representation of sinusoidal signal
Usually, the signals are real-valued. However, the concept of complex signals is a useful tool in
telecommunication. Most of the cases can be handled by using real signals, however, the complex
signals are widely used in spectral analysis.
The following Euler's equations are often needed:
e ± jθ = cos θ ± j sin θ
On the other hand, sine and cosine are given by
e jθ + e − jθ
cos θ =
= Re e jθ
2
e jθ − e − jθ
sin θ =
= Im e jθ
2j
The complex representation of sinusoidal signal (cont.)
The following notations and conventions are used within this course:
1. The spectrum has one variable, the frequency f (Hz) or the angular frequency ω = 2 πf
(radians). f0 , f1 , fi are used for some fixed frequencies.
2. The phase angle is measured with respect to cosine waves or, equivalently with respect to
the positive real axis of the phasor diagram.
3. Amplitude is always positive: − A cos ωt = A cos(ωt ± 180° ) Phase is given in degrees (° )
although radians are usually used.
Two-sided spectrum
One-sided spectrum could be used for real signals. In the following two-sided spectrum is used because
it allows to handle also complex signals.
In the case of real signals, the two-sided spectrum is obtained by using the substitution:
A jφ jω 0t A − jφ − jω 0t
A cos( ω 0 t + φ ) = e e
+ e e
2
2
The two-sided spectrum for the
previous example is shown in the
close figure.
Here, the basis functions are
complex exponential.
Phasor representation
A complex exponential function can be given as phasor which rotates around origin:
Real signal corresponds to the real part of the
phasor:
A cos(ω 0 t + φ ) = Re Ae j (ω 0t + φ)
Phasor representation is used to illustrate
sinusoidal signals and communication signals
consisting of sinusoidals.
Phasor representation (cont.)
The phasor diagram for the two-sided spectrum of sinusoidal consists of two vectors whose phase
and direction of rotation are reversed. The resultant vector is a real signal.
Periodic Signals
The signal v( t ) is periodic if
v( t ± mT0 ) = v( t )
−∞<t <∞
where m is any integer. In this case, the signal can be constructed by combining signal segments of
length T0 :
The length of the periodic signals is infinite, therefore, the signals in the practical system can not be
strictly periodical. However, many finite-length signals in the practical systems correspond very
accurately to the pure periodic signals.
Power and average of the periodic signals
The average of a signal is:
T /2
1
v(t ) = lim
v(t )dt
∫
T →∞ T −T / 2
(arbitrary signal)
1
v(t )dt
∫
T0 T0
(periodic signal)
=
Here the notation ∫ means
T0
t1 +T0
∫ .
t1
The average power of the periodic signal is:
Example: v( t ) = A cos(ω 0 t + φ ) =>
P = | v (t ) | 2 =
v(t ) = 0
1
2
v
(
t
)
dt
∫
T 0 T0
P = A2 / 2
For periodic signals, it is usually assumed that the power is finite, 0 < P < ∞ .
Fourier-series
A periodic signal can be written by using the exponential Fourier series
v(t ) =
∞
∑ cn e
j 2 πnf0t
n =−∞
1
f0 =
T0
1
− j 2πnf 0 t
cn =
v
(
t
)
e
dt
∫
where
T0 T0
j arg c
c
=
c
e
The complex coefficients c can be expressed using the polar form: n
n
n
n
The exponential Fourier series determines the two-sided spectrum for a periodic signal. It consists of the
harmonics (i.e., integer multiples) of the frequency f0 .
cn is the value of the amplitude spectrum at nf0 and arg cn is the corresponding value of the phase
spectrum.
The properties of the line spectrum
1.
All frequencies are integer multiples or harmonics of the fundamental frequency f0
2.
The DC component c0 equals the average value of the signal:
c0 = v ( t )
3.
If v ( t ) is real, then
c − n = cn
arg c− n = − arg cn
which means that the amplitude spectrum has even symmetry and the phase spectrum has odd
symmetry.
Sinc-function
The sinc-function is often needed in the spectral analysis:
sinc λ =
sin πλ
πλ
Example: Rectangular pulse train
The coefficients of the Fourier series can be calculated as follows:
τ /2
T /2
1 0
1
− j 2πnf 0 t
− j 2πnf 0 t
cn =
v
t
e
dt
Ae
dt
=
(
)
T0 −T∫0 / 2
T0 −τ∫/ 2
(
=
A
e − jπnf 0τ − e jπnf 0τ
− j 2πnf 0T0
=
A sin πnf 0τ
T0 πnf 0
=
Aτ
sincnf 0τ
T0
)
Example: Rectangular pulse train (cont.)
Below are the amplitude and phase spectrums in the case where τ / T0 = 1 / 4 . The function sinc fτ can
be recognized from the envelope of the amplitude spectrum. The amplitude of the DC component is
c0 = Aτ / T0 , which can be also easily calculated from the time function.
Gibbs phenomenon
If the periodic signal has a stepwise discontinuity (like in rectangular pulse train), the Fourier series
does not converge at the points of discontinuity. The partial sum converges to the mid-point of the
discontinuity. On each side of discontinuity, v n (t ) has oscillatory overshoot of about 18% independent
of N (the number of terms). As N is increased, the oscillations collapse into nonvanishing spikes.
Gibbs phenomenon has implications for the shapes of the
filters used with real signals. An ideal filter that is shaped
like a rectangular pulse will result in discontinuities in the
spectrum that will lead to distortions in the time signal.
Real applications use window shapes with better timefrequency characteristics, such as Hamming or Hanning
windows.
Also, the signals in the practical systems are always
bandlimited and, thus, do not contain discontinuities.
Therefore, Gibbs phenomenon is not usually a problem in
spectral analysis.
Parseval's theorem
Parseval's theorem relates the average power P of a periodic signal to its Fourier coefficients as follows:
∞
1
2
2
P = ∫ v(t ) dt = ∑ cn
T0 T0
n=-∞
This means that the average power is the sum of the powers of the spectral components.
Therefore, Parseval’s theorem implies superposition of average power.
Fourier transform and continuous spectra
∞
Consider signals whose energy E = ∫ v(t ) dt is finite.
2
−∞
This also means that the signal is concentrated to relatively short time period (time-limited).
The Fourier transform for this kind of energy signal is defined as
V( f ) =
∞
F[v(t )] = ∫ v(t )e
− j 2πft
dt
−∞
V ( f ) is the spectrum of signal v ( t ) .
Nonperiodic signals have continuous spectra.
Periodic signals have line spectra (discontinous spectra).
Fourier transform and continuous spectra
The spectrum has the following properties:
1. V ( f ) is a complex function. V ( f ) is the amplitude spectrum and arg V ( f ) is the phase
spectrum.
∞
2. The value of V at f=0 equals the net area of v( t ) :
V (0) = ∫ v(t )dt
−∞
3. If v( t ) is real => V ( − f ) = V ∗ ( f ) (hermitian symmetry), i.e,
V (− f ) = V ( f )
arg V ( − f ) = − arg V ( f )
The time function v( t ) is obtained from V ( f ) by using the inverse Fourier transform:
∞
v(t ) = F-1[V ( f )] = ∫ V ( f )e j 2πft df
−∞
Example: Rectangular pulse
The notation Π (t / τ ) is used for rectangular pulse. It is defined as
1
Π (t / τ ) = 
0
t <τ /2
t >τ /2
Consider the signal v(t ) = AΠ (t / τ ) . Its Fourier transform is
τ /2
V ( f ) = ∫ Ae
−τ / 2
− j 2πft
Aτ
dt =
sin πfτ = Aτ sinc fτ
πfτ
It can be seen that the spectrum of the rectangular pulse corresponds to the envelope of the spectrum of
the rectangular pulse train (see the previous example).
It can be also noted that most of the spectral content is located in the frequency band of f < 1 / τ . This
means that the spectrum of a narrow pulse is wide.
The Fourier transform of symmetrical signals
V ( f ) = Ve ( f ) + jVo ( f )
The Fourier transform of signal v ( t ) can be expressed in the form:
where
∞
Ve ( f ) = ∫ v (t ) cos ωt dt
−∞
∞
Vo ( f ) = − ∫ v (t ) sin ωt dt
If v ( t ) is real then Ve ( f ) = Re V ( f )
and Vo ( f ) = Im V ( f ) .
If the signal is symmetrical, that is v ( − t ) = v ( t ), then V ( f ) = Ve ( f ) .
If the signal is anti-symmetrical, that is v ( − t ) = − v ( t ), then
V ( f ) = jVo ( f ).
In practice, symmetry depends on the location of the signal with respect
to t = 0 . Even small shift in time destroys the symmetry. However, in
many cases this kind of signal can be located such that it is symmetrical
with respect to t = 0 .
−∞
The Fourier transform of causal signals
The signal is causal if
v (t ) = 0
t<0
∞
For causal signals, the integral in the Fourier transform can be calculated in the range ∫0 . In
this case, the Fourier transform is a special case of the Laplace transform.
The causality is unnecessarily restrictive assumption for telecommunication signals, and
therefore, it is of little consequence.
Rayleigh's energy theorem
Rayleigh's energy theorem is similar to Parseval's theorem (* is the complex conjugate)
∞
∞
∞
−∞
−∞
E = ∫ v (t ) dt = ∫ V ( f )V ∗ ( f )df = ∫ V ( f ) 2 df
2
−∞
Thus, the energy of the signal can be calculated by integrating the square of the amplitude spectra.
Example:
The total energy of the rectangular pulse AΠ( t / τ ) is E = A τ .
2
The energy in the frequency band f < 1 / τ ) is
1/ τ
1/τ
2
2
2
V
(
f
)
df
=
(
A
τ
)
sinc
f
τ
df
=
0
.
92
A
τ
∫
∫
−1/ τ
2
−1/ τ
This is 92% of the total energy.
Duality
If v ( t ) and V ( f ) are Fourier transform pairs, then the duality transform is
F[V (t )] = ν (- f ) .
This result can be easily obtained by changing the variables in the integrals of the Fourier transform and
its inverse transform.
This property can be used to derive the duality transform pairs by using the known transform pairs.
Example:
Rectangular pulse
t
v(t ) = AΠ ( )
τ
V ( f ) = Aτ sinc fτ
Sinc-pulse
z (t ) = Asinc 2Wt
A
f
)
Z( f ) =
Π(
2W
2W
The spectrum of the sinc-function is the bandlimited rectangular function.
Reversly, the spectrum of a rectangular function is the sinc-function.
Transform calculations
The integral is usually difficult to be calculated analytically. Other methods:
1. Transform tables and transform theorems (duality, frequency and time shift, etc.)
2. Approximation. If z%( t ) approximates z ( t ) then
∞
∞
−∞
−∞
2
2
~
~
Z
(
f
)
−
Z
(
f
)
df
=
z
(
t
)
−
z
(
t
)
dt
∫
∫
This implies that the frequency-domain approximation can be improved by improving the timedomain approximation.
3. Numerical methods: Discrete Fourier transform (DFT)
Can be implemented using, e.g., Matlab.
The difference between the continuous Fourier transform and DFT has to be known (e.g., the
spectrum of a discrete signal is periodical!)
Time and frequency relations
Transform theorems are very important when calculating Fourier transforms (e.g., using
tables). They are needed when we derive some general results for the spectra.
1.
Superposition
2.
Time delay
a1v1 ( t ) + a2 v2 ( t ) ↔ a1V1 ( f ) + a2 V2 ( f )
v (t − t d )↔V ( f )e
− j 2πft d
Delay has some effect only to the phase spectra, amplitude spectra are unchanged.
3.
Time scale
v(αt )↔
f
V 
α α 
1
α ≠0
For example, the spectra is spread out when the signal is shrunk.
Time and frequency relations (cont.)
4.
Differentiation
d
v( t ) ↔ j 2 πfV ( f )
dt
Differentiation emphasizes high frequencies.
5.
Integration
∞
1
V( f )
j 2πf
−∞
Integration emphasizes low frequencies.
∫ v(λ )dλ ↔
6.
Frequency translation and modulation
v(t )e j 2πf t ↔V ( f − f c )
c
Complex modulation, that is multiplication by e j 2 πfct , shifts the spectra by + fc . This is the
fundamental idea behind the linear modulation methods.
Time and frequency relations (cont.)
In the above frequency transformation, the negative components of the spectra move to the
positive frequencies, and thus, the bandwidth is doubled. Example of frequency translation:
Bandwidth W
Bandwidth 2W
The frequency shifted signal is complex even if the original signal is real.
Time and frequency relations (cont.)
The following result can be derived for real signals:
jφ
− jφ
e
e
v (t ) cos(2πf ct + φ )↔ V ( f − f c ) +
V ( f + fc )
2
2
7.
Convolution
v∗ w ( t )
↔ V ( f )W ( f )
v(t )w(t ) ↔ V ∗ W ( f )
Convolution in time domain is equivalent to the multiplication in frequency domain.
Multiplication in time domain is equivalent to the convolution in frequency domain.
Convolution
The convolution between two signal v( t ) and w ( t ) is denoted by v∗ w ( t ) and it is determined by
∞
v ∗ w(t ) = ∫ v(λ ) w(t − λ )dλ
−∞
Convolution has the following properties:
- Commutativity, associativity and distributivity
v∗ w = w∗v
v ∗ ( w ∗ z ) = (v ∗ w) ∗ z
v ∗ ( w + z ) = (v ∗ w) + (v ∗ z )
Convolution is an essential concept when analyzing linear continuous-time systems.
Convolution (cont.)
The figure to the right illustrates the calculation
of convolution.
The result of convolution:
Example of signal spectrum for a train of pulses
Left figure:
whe re v (t ) = AΠ (t / τ )
z a ( t ) = v ( t − td ) − v ( t − td − T )
By using superposition and time shift, we can write:
Z a ( f ) = V ( f )e − j 2πft − V ( f )e − j 2πf ( t +T ) = V ( f )e − j 2πft [e jπfT − e − jπfT ]
d
0
d
= ( Aτ sinc fτ )( j 2 sin(πfT )e − j 2πft )
0
where t0 = td + T / 2 .
Example of signal spectrum for a train of pulses (Cont.)
Similarly, for the figure at the right, t0 = 0 and T = τ , thus
 t +τ / 2 
 t −τ / 2 
zb (t ) = AΠ 
 − AΠ 

 τ 
 τ 
2
(
)(
)
Z
(
f
)
=
A
τ
sinc
f
τ
j
2
sin
π
f
τ
=
(
j
2
π
f
τ
)
A
τ
sinc
fτ
and the spectra is b
The time function is asymmetrical and the spectrum has only the imaginary part.
Example of the signal spectrum for a truncated sinusoid
RF-pulse:
t
z (t ) = AΠ   cos ω c t
τ 
The Fourier transform is:
Aτ
Aτ
Z( f ) =
sinc ( f − fc ) τ +
sinc ( f + fc ) τ
2
2
The spectra of a periodic (infinite long) cosine signal would consist only impulses at the
frequencies fc and − fc . Because of the truncation, the spectrum spreads out around these
frequencies.
Impulses
Until now: a clear distinction between line spectra (that represent periodic signals) and
continuous spectra (that represent non-periodic signals).
Sometimes, a signal has both periodic and non-periodic terms => we introduce the concept of
impulses in frequency domain for the representation of discrete frequency components.
Useful tool: Dirac delta function (unit impulse).
The unit impulse or the Dirac delta function δ (t ) is defined as:
t2
t1 < 0 < t 2
v(0)
v(t )δ (t )dt = 
otherwise
 0
t1
Here v( t ) is any ordinary function that is continuous at t = 0 . The impulse is mathematically
defined only when it is inside the integral.
∫
∞
ε
−∞
−ε
δ (t )dt = ∫ δ (t )dt = 1
∫
From the equation above, we can derive
δ (t ) = 0
when t ≠ 0
Impulses
Graphical representation of the impulse: Aδ (t − td )
Although an impulse does not exist physically, there are many conventional functions that
have all the properties of the impulse as some parameter goes to zero. For example, a
rectangular pulse having the amplitude of 1 / ε and width of ε approaches an impulse
waveform when ε approaches zero.
The impulse has the properties:
v(t ) ∗ δ (t − td ) = v(t − td )
∞
∫ v(t )δ (t − t )dt = v(t )
d
d
−∞
Same property holds for the sinc function.
Impulse in the frequency domain
Frequency-domain impulse corresponds to the spectra of constant and sinusoidal signals.
The following transform pairs can be given:
A ↔ Aδ ( f )
Ae jω c t
↔
Aδ ( f − fc )
A cos( ω c t + φ ) ↔
Ae jφ
Ae − jφ
δ ( f − fc ) +
δ ( f + fc )
2
2
The spectra of the sinusoidal signals contains two impulses:
Impulse in the frequency domain
∞
j 2 πnf0 t
v
(
t
)
=
c
e
∑ n
If the Fourier series of the periodic signal is:
n =−∞
∞
Then its Fourier transform is
V ( f ) = ∑ cnδ ( f − nf 0 )
n = −∞
The above relationship connects the line spectrum of the periodic signals to the spectrum
determined by the Fourier transform.
Impulses in the time domain
The following transform pairs can be derived:
Aδ ( t ) ↔
A
Aδ ( t − td ) ↔
Ae − jωtd
The spectrum of the time-domain impulse contains all frequencies in equal proportion.
This is, of course, physically impossible. However, there is some phenomenon where this
model is valid up to very high frequencies.
Step and sign-functions
a) Step function is defined by
1 t > 0
u (t ) = 
0 t < 0
b) Sign-function is defined by
1 t >0
sgn t = 
− 1 t < 0
The time-Fourier transform pairs for these functions
a) u( t )
↔
1
δ( f )
+
j 2 πf
2
and b)
sgn t
↔
1
jπ f