Signals and Spectra

Chapter 2: Signals and Spectra

in Communication Systems  A generic sinusoidal
signal
v(t )  A cos(0t   ); 0  2f 0
 Phasor representation

January 2024
Lectured by Prof. Dr. Thuong Le-Tien  Frequency domain
representation

Amplitude
 Rotating phasors A
 Frequency plots f0
Slides with references from HUT Finland, La Hore uni.,  Amplitude A f0 f
Mc. Graw Hill Co., A.B. Carlson’s “Communication  Phase 0t   

Phase
Systems”, and Leon W.Couch “Digital and Analog
Communication Systems” books f0 f
1 2

 Two sided spectra can be seen from

Periodic Signals
 This represents two rotating phasors
 Amplitude and phase spectrum (two sided)  A signal x p(t ) is periodic if there exists T
such that x p(t) = x p(t + T)
 Smallest such T is called fundamental
period T0
 Any integer multiple of T0 is also a period

T0

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 Normalized Power
Average signal and Power  In the concept of normalized power, R is
assumed to be 1Ω, although it may be another
 Average signal value in the actual circuit.
 Another way of expressing this concept is to say
that the power is given on a per-ohm basis.
 It can also be realized that the square root of the
normalized power is the rms value.
 For periodic signals
Definition. The average normalized power is given by:
Where s(t) is the voltage or current waveform
 Average power T /2
1
P  s (t )  lim 
2
s 2 (t )dt
T  T
T / 2
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Decibel Decibel Gain

 A base 10 logarithmic measure of power ratios.
 The ratio of the power level at the output of a circuit
compared with that at the input is often specified by
 If resistive loads are involved,
the decibel gain instead of the actual ratio.
 Decibel measure can be defined in 3 ways
 Decibel Gain
 Decibel signal-to-noise ratio (SNR in dB) Definition of dB may be reduced to,
 Mili-watt Decibel or dBm
 Definition: Decibel Gain
The decibel gain of a circuit is:
or

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Decibel signal-to-noise ratio (SNR) Decibel with mili watt reference (dBm)
 Definition. The decibel power level with respect to 1 mW
 Definition. The decibel signal-to-noise ratio
(SNR) is:

= 30 + 10 log (Actual Power Level (watts)

Where, Signal Power (S) =  Here the “m” in the dBm denotes a milliwatt reference.
 When a 1-W reference level is used, the decibel level is
denoted dBW;
And, Noise Power (N) =  when a 1-kW reference level is used, the decibel level
is denoted dBk.
E.g.: If an antenna receives a signal power of 0.3W, what is the
received power level in dBm?
dBm = 30 + 10xlog(0.3) = 30 + 10x(-0.523)3 = 24.77 dBm

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Fourier Series
Fourier Series Representation
 Projection of periodic signals onto basis
functions
 Periodic signal is a weighted sum of these basis
functions
 Exponentials are used as basis functions for DC component:
writing Fourier series
Any periodic signal can be expressed as a
1
 f0= T0 (fundamental frequency)
sum of infinite number of exponentials (or Line spectra at frequencies that are integer
sinusoids for real signals) multiple of fundamental frequency

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Fourier series example: Fourier Series: Example

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Three major properties of V(f)

Fourier Transform
 Back to the Fourier series:

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Rectangular pulse spectrum
V(ƒ) = A sinc ƒ

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Convolution
 The convolution of a waveform w1(t) with a waveform
w2(t) to produce a third waveform w3(t) which is

Evaluation of the integral involves 3 steps.

• Time reversal of w2 to obtain w2(-λ),
• Time shifting of w2 by t seconds to obtain w2(-(λ-t)),
and
• Multiplying this result by w1 to form the integrand
w1(λ)w2(-(λ-t)).

Note: we denote a signal s(t) as a waveform w(t)

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Example for Convolution Power Spectral Density (PSD)
 T 
t 2   We define the truncated version of the waveform by:
w1 (t )    
 T 
 
t
-
w 2 (t)=e T u (t )

For 0< t < T • The average normalized power:

For t > T
• Using Parseval’s theorem to calculate power from the
frequency domain

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Autocorrelation Function
 Definition: The Power Spectral Density (PSD) for a  Definition: The autocorrelation of a real (physical)
deterministic power waveform is waveform is

• Wiener-Khintchine Theorem: PSD and the autocorrelation function are Fourier

• where wT(t) ↔ WT(f) and Pw(f) has units of watts per hertz. transform pairs;

• The PSD is always a real nonnegative function of frequency.

The PSD can be evaluated by either of the following two methods:
• PSD is not sensitive to the phase spectrum of w(t) 1. Direct method: by using the definition,
• The normalized average power is 2. Indirect method: by first evaluating the autocorrelation function and
then taking the Fourier transform:

Pw(f)= ℑ [Rw(τ) ]
• This means the area under the PSD function is the normalized
• The average power can be obtained by any of the four techniques.
average power.

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 Normalized Power Power Spectral Density for Periodic Waveforms
Theorem: For a periodic waveform, the power spectral
Theorem: For a periodic waveform w(t), the
density (PSD) is given by
normalized power is given by:

where T0 = 1/f0 is the period of the waveform and

where the {cn} are the complex Fourier coefficients for the waveform. {cn} are the corresponding Fourier coefficients for the waveform.

Proof: For periodic w(t), the Fourier series representation is valid over all time
and one may evaluate the normalized power:

PSD is the FT of the

Autocorrelation
function

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Power Spectral Density for a Square Wave Noise in communication systems

• The PSD for the periodic square wave will be found.  Thermal noise is described by a zero-mean Gaussian random
• Because the waveform is periodic, FS coefficients can be used to process, n(t).
evaluate the PSD. Consequently this problem becomes one of  Its PSD is flat, hence, it is called white noise.
evaluating the FS coefficients.
[w/Hz]

Power spectral
density

Autocorrelation
function

Probability density function

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Signal transmission through linear systems  Ideal filters:

Input Output
Non-causal!
Linear system Low-pass

 Deterministic signals:
 Random signals:
Band-pass High-pass

 Ideal distortion less transmission:

All the frequency components of the signal not

 Realizable filters:
only arrive with an identical time delay, but also
RC filters Butterworth filter
are amplified or attenuated equally.


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Bandwidth of signal Bandwidth of signal …

 Different definition of bandwidth:
 Baseband versus bandpass: a) Half-power bandwidth a) Fractional power containment bandwidth
b) Noise equivalent bandwidth b) Bounded power spectral density
Baseband Bandpass c) Null-to-null bandwidth c) Absolute bandwidth
signal signal
Local oscillator

(a)
 Bandwidth dilemma: (b)
 Bandlimited signals are not realizable! (c)
(d)
 Realizable signals have infinite bandwidth!
(e)50dB
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 Power Transfer Function
 Derive the relationship between the power spectral density
(PSD) at the input, Px(f), and that at the output, Py(f) , of a linear
time-invariant network.
Using the definition of PSD

PSD of the output is

Using transfer function

in a formal sense, we obtain

Thus, the power transfer

function of the network is

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