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Signals and Systems Review: Fourier Transform & Analysis

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Slide 26
Topic 2: Review of Signals and Linear Systems
 At the end of this chapter, you will be able to:
 Signals and systems.
 Signal energy and power.
 Signal classification into: continuous/discrete, analog/digital, periodic/aperiodic…
 Time shifting, time scaling, and reversal of time/signal.
 Unit impulse and unit step functions.
 Signal orthogonality.
 Signal correlation, cross-correlation, and auto-correlation.
 Generalized, trigonometric, exponential, and discrete Fourier series.
 The Fourier transform.
 Characteristics and properties of the Fourier transform.
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Signals and Systems
Slide 27
 A signal is a set of information or data represented as a function of time or
space.
 A system is an entity that processes a set of input signals to produce a set of
output signals.
 Systems are used to process signals to modify them or extract information from
them.
 Systems may:
 Contain physical components (electrical circuits).
 Be software algorithms that process digital sequences.
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Signal Energy and Power
Slide 28
 The energy of a voltage signal 𝒈 𝒕 is defined as the area under its square:
+∞
𝑬𝒈 = න
𝒈 𝒕
𝟐 𝒅𝒕
−∞
 The notation 𝒈 𝒕
refers to the magnitude of the complex valued 𝒈 𝒕 .
 The value of 𝑬𝒈 is indicative of the energy that can be extracted from the
signal.
 The energy 𝑬𝒈 may be infinite. It can only be finite if the signal amplitude → 𝟎
as → ∞.
 A more meaningful measure of the strength of 𝒈 𝒕 if the energy is infinite is its
power, which is defined as the time average of energy:
𝟏 +𝑻/𝟐
𝑷𝒈 = 𝐥𝐢𝐦 න
𝒈 𝒕 𝟐 𝒅𝒕
𝑻→∞ 𝑻 −𝑻/𝟐
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Slide 29
Dissipated Energy
 The signal energy 𝑬𝒈 can be interpreted as the energy dissipated by a 𝟏𝛀
resistor when the voltage 𝒈 𝒕 is applied across it.
 If the value of the resistor is 𝑹𝛀, the energy dissipated is:
+∞
𝑬𝒈
𝒈 𝒕 𝟐
𝑬=න
𝒅𝒕 =
𝑹
𝑹
−∞
 Hence, the energy of a signal can be thought of as an indication of its
capability and not the actual energy dissipated.
 The energy of the signal is useful when assessing the quality of a
communication system. For instance, if 𝒈 𝒕 is the source message and 𝒛 𝒕 is
the reconstructed message, the communication error can be defined as:
𝒆 𝒕 =𝒈 𝒕 −𝒛 𝒕
 The energy or power of 𝒆 𝒕 quantifies the quality of the system.
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Example 2.1
Slide 30
 We want to determine suitable measures for the signals below.
 The first signal 𝒈 𝒕 → 𝟎 as 𝒕 → ∞. Hence, energy is suitable:
+∞
𝑬𝒈 = න
𝟎
𝒈 𝒕
𝟐
+∞
𝒅𝒕 = න 𝟐𝟐 𝒅𝒕 + න
−∞
−𝟏
𝟒𝒆−𝒕 𝒅𝒕 = 𝟖𝑽𝟐 𝒔
𝟎
 The second signal’s energy is infinite and thus power is more suitable as a
measure.
 We have:
𝑻
+𝟐
𝟏
න 𝒈 𝒕
𝑻→∞ 𝑻 −𝑻
𝑷𝒈 = 𝐥𝐢𝐦
𝟐 𝒅𝒕
𝟐
𝟏
𝟏
𝟏
න 𝒕𝟐 𝒅𝒕 =
𝑻→∞ 𝟐 −𝟏
𝟑
= 𝐥𝐢𝐦
 The RMS value of this signal is 𝟏ൗ 𝟑.
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Example 2.2
Slide 31
 Determine the power and RMS value of:
𝒈𝟏 𝒕 = 𝑪 𝐜𝐨𝐬 𝝎𝟎 𝒕 + 𝝋
𝒈𝟐 𝒕 = 𝑫𝒆𝒋𝝎𝟎𝒕
 First, the period of 𝒈𝟏 𝒕 is 𝑻𝟎 = 𝟐𝝅Τ𝝎𝟎. Hence,
𝑻
+
𝟐
𝟏
න 𝑪𝟐 𝒄𝒐𝒔𝟐 𝝎𝟎 𝒕 + 𝝋 𝒅𝒕
𝑻→∞ 𝑻 −𝑻
𝑷𝒈 = 𝐥𝐢𝐦
𝟐
𝑻
𝟐 𝑪𝟐
𝟏
න
𝟏 + 𝐜𝐨𝐬 𝟐𝝎𝟎 𝒕 + 𝟐𝝋 𝒅𝒕
𝑻→∞ 𝑻 −𝑻 𝟐
= 𝐥𝐢𝐦
𝟐
𝑻
𝑪𝟐 𝟐
𝑻
𝟐
𝑪𝟐
𝑪𝟐
= 𝐥𝐢𝐦
න 𝒅𝒕 + 𝐥𝐢𝐦
න 𝐜𝐨𝐬 𝟐𝝎𝟎 𝒕 + 𝟐𝝋 𝒅𝒕 =
𝑻
𝑻→∞ 𝟐𝑻 −𝑻
𝑻→∞ 𝟐𝑻
𝟐
−
𝟐
𝑪𝟐ൗ
𝟐
𝟐
𝒂𝒕 𝒎𝒐𝒔𝒕 𝒕𝒉𝒆 𝒂𝒓𝒆𝒂 𝒐𝒇 𝒂 𝒉𝒂𝒍𝒇 𝒄𝒚𝒄𝒍𝒆
𝟎
 The RMS value is simply the square root of the power.
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Example 2.2
Slide 32
 The second example is complex. The magnitude squared is given by:
𝒈 𝒕
𝟐 =
𝑫𝒆𝒋𝝎𝟎𝒕
𝟐
= 𝑫 𝟐 𝒆𝒋𝝎𝟎𝒕
𝟐
= 𝑫𝟐
 Hence,
𝟐 𝑻
𝟏 𝑻
𝑫
𝟐
𝑷𝒈 = 𝐥𝐢𝐦 න 𝑫𝒆𝒋𝝎𝟎𝒕 𝒅𝒕 = 𝐥𝐢𝐦
න 𝒅𝒕 = 𝑫 𝟐
𝑻→∞ 𝑻 𝟎
𝑻→∞ 𝑻
𝟎
 Thus, the RMS value is simply 𝑫 .
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Slide 33
Continuous vs. Discrete-Time Signals
 Signals are classified with respect to time into:
 Continuous time: specified at every value of time 𝒕.
 Discrete time: specified at discrete values of 𝒕.
 An example is shown below for the 𝒔𝒊𝒏𝒄 𝒙 function:
 The continuous signal is
denoted by 𝒈 𝒕 .
 The discrete signal is
denoted by 𝒈 𝒏 with the
index 𝒏 = 𝟎, 𝟏, 𝟐, 𝟑, 𝟒, ….
 Note that discretization is
only the first step in the
analog to digital
conversion process.
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Slide 34
Analog vs. Digital Signals
 From another perspective, signals can be classified according to their
amplitude into:
 Analog: where the amplitude can take any value in a continuous range.
 Digital: where the amplitude can only take one of a finite number of values.
 In general, when we move from analog to digital, we go through two steps:
 Discretization: where the signal is sampled at discrete time instants.
 Quantization: where the samples are approximated to the nearest interval midpoint
as we saw in the first chapter.
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Slide 35
Periodic vs. Aperiodic Signals
 A signal 𝒈 𝒕 is said to be periodic if there exists a positive constant 𝑻𝟎 such
that for all 𝒕:
𝒈 𝒕 = 𝒈 𝒕 + 𝑻𝟎
 The smallest value of 𝑻𝟎 is called the period of 𝒈 𝒕 .
 If a signal is not periodic, it is said to be aperiodic.
 Note that a periodic signal must start at 𝒕 = −∞.
 If 𝑻 is the period, then it is also true that: 𝒈 𝒕 = 𝒈 𝒕 + 𝒎𝑻𝟎 for any integer 𝒎.
 A periodic signal can be generated by simply taking any segment of duration
𝑻 and repeating it.
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Slide 36
Time-Shifting and Scaling
 Adding a positive constant 𝑻 to the time variable 𝒕 yields 𝒈 𝒕 + 𝑻 , which is a
shifted version to the left (advanced).
 Similarly, 𝒈 𝒕 − 𝑻 is shifted to right (delayed).
 Time scaling is the compression or expansion of the signal in time.
 Time scaling is performed by multiplying the time variable by a constant 𝒂
yielding 𝒈 𝒂𝒕 :
 If 𝒂 > 𝟏: the signal is compressed.
 If 𝒂 < 𝟏: the signal is expanded.
 In the plot on the right, the signal
𝒈 𝒕 is compressed by a factor of 𝟐
and then expanded by the same
factor.
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Slide 37
Reversal of Time and Signal
 Inverting the time variable 𝒕 causes the signal to flip with respect to the 𝒙-axis.
 Inverting the signal 𝒈 𝒕 causes it to flip with respect to the 𝒚-axis.
 Recall that a function is said to be:
 Odd: if it is symmetric about the origin, that is: 𝒈 −𝒕 = −𝒈 𝒕 .
 Even: if it is symmetric about the 𝒚-axis, that is: 𝒈 −𝒕 = 𝒈 𝒕 .
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Slide 38
Unit Impulse Function
 The impulse function is defined as:
𝜹 𝒕 = 𝟎 for 𝒕 ≠ 𝟎
∞
න 𝜹 𝒕 𝒅𝒕 = 𝟏
−∞
 It can be thought of as a tall rectangle with an extremely
short width 𝝐.
 As 𝒕 → 𝟎, 𝜹 𝒕 is equal to a very high value 𝟏Τ𝝐.
 Note that the function 𝜹 𝒕 is undefined at 𝒕 = 𝟎.
 Also note that 𝜹 𝒕 is not an ordinary function because it is not defined for all 𝒕
(only for 𝒕 ≠ 𝟎).
 It is said to be a generalized function.
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Slide 39
Sampling Using the Impulse Function
 Provided a function 𝝋 𝒕 is continuous at 𝒕 = 𝟎 and 𝒕 = 𝑻, we see that:
𝜹 𝒕 𝝋 𝒕 =𝜹 𝒕 𝝋 𝟎
𝜹 𝒕−𝑻 𝝋 𝒕 =𝜹 𝒕−𝑻 𝝋 𝑻
 This becomes useful in sampling.
 We have:
∞
∞
න 𝝋 𝒕 𝜹 𝒕 𝒅𝒕 = 𝝋 𝟎 න 𝜹 𝒕 𝒅𝒕 = 𝝋 𝟎
−∞
∞
−∞
න 𝝋 𝒕 𝜹 𝒕 − 𝑻 𝒅𝒕 = 𝝋 𝑻
−∞
 Hence, the signal 𝝋 𝒕 can be sampled by multiplying it with delayed impulses
assuming that 𝝋 𝒕 is continuous at the impulse locations.
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The Unit Step Function
Slide 40
 The unit step is another important function in communications:
𝟏 𝒕≥𝟎
𝒖 𝒕 =ቊ
𝟎 𝒕<𝟎
 When 𝒖 𝒕 is multiplied by a signal 𝒈 𝒕 , the resulting signal becomes zero for all
𝒕 < 𝟎. In this case, it is said to be causal.
 For instance:
 The signal 𝒈 𝒕 = 𝒆−𝒂𝒕 is non-causal.
 The signal 𝒖 𝒕 𝒆−𝒂𝒕 is causal.
 The unit function can be defined as:
𝒕
𝒖 𝒕 = න 𝜹 𝝉 𝒅𝝉
→
𝜹 𝒕 =
−∞
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Slide 41
Orthogonality and Correlation
 Two complex signals 𝒙𝟏 𝒕 and 𝒙𝟐 𝒕 are orthogonal over the interval 𝒕𝟏 , 𝒕𝟐 if:
𝒕𝟐
න
𝒕𝟏
𝒙𝟏 𝒕 𝒙∗𝟐 𝒕 𝒅𝒕 = 𝟎
𝒕𝟐
or
න 𝒙𝟏∗ 𝒕 𝒙𝟐 𝒕 𝒅𝒕 = 𝟎
𝒕𝟏
 The correlation coefficient 𝒄𝒏 measures the similarity between two real signals:
+∞
𝟏
𝒄𝒏 =
න 𝒈 𝒕 𝒙 𝒕 𝒅𝒕
𝑬𝒈 𝑬𝒙 −∞
 For complex signals, we simply have:
+∞
𝟏
𝒄𝒏 =
න 𝒈 𝒕 𝒙∗ 𝒕 𝒅𝒕
𝑬𝒈 𝑬𝒙 −∞
 Note that −𝟏 ≤ 𝒄𝒏 ≤ 𝟏. When:
 𝒄𝒏 = 𝟏: This means that the two signals are identical.
 𝒄𝒏 = −𝟏: This means that the two signals are inverted versions of one another.
 𝒄𝒏 = 𝟎: This means that the two signals are orthogonal.
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Example 2.6
Slide 42
 Calculate the correlation of 𝒙 𝒕 with each of the three
signals.
 We have:
+∞
𝑬𝒙 = න
𝟓
𝒙 𝒕
𝟐 𝒅𝒕 = න 𝒅𝒕 = 𝟓
−∞
𝟎
 For 𝒈𝟏 𝒕 :
𝟓
𝑬𝒈𝟏 = න 𝒅𝒕 = 𝟓
𝒄𝒏 =
𝟏
𝟎
+∞
න 𝒈 𝒕 𝒙 𝒕 𝒅𝒕
𝑬𝒈 𝑬𝒙 −∞
𝟓
𝟏
=
න 𝒅𝒕
𝟓×𝟓 𝟎
=𝟏
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Example 2.6
Slide 43
 For 𝒈𝟐 𝒕 :
𝟓
𝑬𝒈𝟐
= න 𝒆−𝟐𝒕Τ𝟓 𝒅𝒕 = −𝟐. 𝟓𝒆−𝟐𝒕Τ𝟓 ቚ
𝟎
𝒄𝒏 =
 For 𝒈𝟑 𝒕 :
𝟏
𝟎
= 𝟐. 𝟏𝟔𝟏𝟕
𝟓
න 𝒆−𝒕Τ𝟓 𝒅𝒕 = 𝟎. 𝟗𝟔𝟏𝟒
𝟓 × 𝟐. 𝟏𝟔𝟏𝟕 𝟎
𝟓
𝑬𝒈𝟑
𝟓
= න 𝐬𝐢𝐧𝟐 𝟐𝝅𝒕
𝟎
𝟏 𝟓
𝒅𝒕 = න 𝟏 − 𝐜𝐨𝐬 𝟒𝝅𝒕 𝒅𝒕
𝟐 𝟎
𝟓 𝟏 𝟓
= − න 𝐜𝐨𝐬 𝟒𝝅𝒕 𝒅𝒕 = 𝟐. 𝟓
𝟐 𝟐 𝟎
𝒄𝒏 =
𝟏
𝟓
න 𝐬𝐢𝐧 𝟐𝝅𝒕 𝒅𝒕 = 𝟎
𝟓 × 𝟐. 𝟓 𝟎
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Slide 44
Cross and Auto-Correlation
 The correlation coefficient seen earlier is insensitive to delays in the signals.
 Another measure is cross-correlation, which includes a time delay:
+∞
𝝍𝒈𝒙 𝝉 = න
𝒈 𝒕 𝒙 𝒕 + 𝝉 𝒅𝒕
−∞
 This measures the similarity between 𝒈 𝒕 and 𝒙 𝒕 + 𝝉 as a function of the
delay 𝝉.
 The correlation of a signal with itself is called auto-correlation:
+∞
𝝍𝒈 𝝉 = න
𝒈 𝒕 𝒈 𝒕 + 𝝉 𝒅𝒕
−∞
 Auto-correlation gives us valuable spectral information about the signal.
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Slide 45
Generalized Fourier Series
 An orthogonal signal set 𝒙𝟏 𝒕 , 𝒙𝟐 𝒕 , … , 𝒙𝑵 𝒕
over an interval 𝒕𝟏 , 𝒕𝟐 satisfies:
+∞
𝟎 𝒎≠𝒏
∗
න 𝒙𝒎 𝒕 𝒙𝒏 𝒕 𝒅𝒕 = ቊ
𝑬𝒏 𝒎 = 𝒏
−∞
 If for all 𝒏, 𝑬𝒏 = 𝟏, then the set is said to be orthonormal. Normalization can be
achieved by simply dividing 𝒙𝒏 𝒕 by 𝑬𝒏 for all 𝒏.
 The generalized Fourier series of a signal 𝒈 𝒕 over the interval 𝒕𝟏 , 𝒕𝟐 is:
∞
𝒈 𝒕 = ෍ 𝒄𝒏 𝒙𝒏 𝒕
𝒏=𝟏
 The set 𝒙𝒏 𝒕 is called the set of basis signals. Note that the equality here
means that the error energy approaches zero.
 Parseval’s theorem states that:
𝑬𝒈 = ෍ 𝒄𝟐𝒏 𝑬𝒏
𝒏
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Slide 46
Trigonometric Fourier Series
 The trigonometric Fourier series is of the form:
∞
𝒈 𝒕 = 𝒂𝟎 + ෍ 𝒂𝒏 𝐜𝐨𝐬 𝒏𝝎𝟎 𝒕 + 𝒃𝒏 𝐬𝐢𝐧 𝒏𝝎𝟎 𝒕
𝒕𝟏 ≤ 𝒕 ≤ 𝒕𝟏 + 𝑻𝟎
𝒏=𝟏
 With 𝝎𝟎 = 𝟐𝝅ൗ𝑻𝟎 and for 𝒏 = 𝟏, 𝟐, 𝟑, …:
𝟏 𝒕𝟏 +𝑻𝟎
𝒂𝟎 =
න
𝒈 𝒕 𝒅𝒕
𝑻𝟎 𝒕𝟏
𝟐 𝒕𝟏 +𝑻𝟎
𝟐 𝒕𝟏 +𝑻𝟎
𝒂𝒏 =
න
𝒈 𝒕 𝐜𝐨𝐬 𝒏𝝎𝟎 𝒕 𝒅𝒕
𝒃𝒏 =
න
𝒈 𝒕 𝐬𝐢𝐧 𝒏𝝎𝟎 𝒕 𝒅𝒕
𝑻𝟎 𝒕𝟏
𝑻𝟎 𝒕𝟏
 The compact form is:
∞
𝒈 𝒕 = 𝑪𝟎 + ෍ 𝑪𝒏 𝐜𝐨𝐬 𝒏𝝎𝟎 𝒕 + 𝜽𝒏
𝒕𝟏 ≤ 𝒕 ≤ 𝒕𝟏 + 𝑻𝟎
𝒏=𝟏
𝑪𝟎 = 𝒂𝟎
𝑪𝒏 =
𝒂𝟐𝒏 + 𝒃𝟐𝒏
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Example 2.7
Slide 47
 Find the compact trigonometric
Fourier series for the exponential 𝒈 𝒕
= 𝒆−𝒕Τ𝟐 over the interval 𝟎, 𝝅 .
 We have 𝝎𝟎 = 𝟐𝝅ൗ𝑻𝟎 = 𝟐𝐫𝐚𝐝/𝒔.
 Therefore,
𝟏 𝝅 −𝒕Τ𝟐
−𝟐 −𝒕Τ𝟐 𝝅
𝒂𝟎 = න 𝒆
𝒅𝒕 =
𝒆
ቚ = 𝟎. 𝟓𝟎𝟒𝟑
𝝅 𝟎
𝝅
𝟎
𝟐 𝝅 −𝒕Τ𝟐
𝟐
𝒂𝒏 = න 𝒆
𝐜𝐨𝐬 𝟐𝒏𝒕 𝒅𝒕 = 𝟎. 𝟓𝟎𝟒𝟑
𝝅 𝟎
𝟏 + 𝟏𝟔𝒏𝟐
𝟐 𝝅 −𝒕Τ𝟐
𝟖𝒏
𝒃𝒏 = න 𝒆
𝐬𝐢𝐧 𝟐𝒏𝒕 𝒅𝒕 = 𝟎. 𝟓𝟎𝟒𝟑
𝝅 𝟎
𝟏 + 𝟏𝟔𝒏𝟐
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Example 2.7
Slide 48
 The trigonometric form can be written as:
∞
𝟐
𝒈 𝒕 = 𝟎. 𝟓𝟎𝟒𝟑 𝟏 + ෍
𝐜𝐨𝐬 𝟐𝒏𝒕 + 𝟒𝒏 𝐬𝐢𝐧 𝟐𝒏𝒕
𝟐
𝟏 + 𝟏𝟔𝒏
𝟎≤𝒕≤𝝅
𝒏=𝟏
 We have 𝑪𝟎 = 𝒂𝟎 = 𝟎. 𝟓𝟎𝟒𝟑 and:
𝑪𝒏 = 𝟎. 𝟓𝟎𝟒𝟑
𝟐
𝟏 + 𝟏𝟔𝒏𝟐
𝟐
𝟖𝒏
+
𝟏 + 𝟏𝟔𝒏𝟐
−𝟏
𝜽𝒏 = 𝐭𝐚𝐧
 The compact form is:
∞
𝒈 𝒕 = 𝟎. 𝟓𝟎𝟒𝟑 𝟏 + ෍
𝒏=𝟏
𝟐
=
𝟔𝟒𝒏𝟐 + 𝟒
= 𝟎. 𝟓𝟎𝟒𝟑
𝟏 + 𝟏𝟔𝒏𝟐 𝟐
𝟐
𝟏 + 𝟏𝟔𝒏𝟐
−𝒃𝒏
= 𝐭𝐚𝐧−𝟏 −𝟒𝒏 = − 𝐭𝐚𝐧−𝟏 𝟒𝒏
𝒂𝒏
𝟐
𝟏 + 𝟏𝟔𝒏𝟐
𝐜𝐨𝐬 𝟐𝒏𝒕 − 𝐭𝐚𝐧−𝟏 𝟒𝒏
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Slide 49
Characteristics of the Fourier Series
 The compact Fourier series is expressed as the sum of sinusoids with
frequencies 𝝎𝒏 = 𝒏𝝎𝟎 , amplitudes 𝑪𝒏 , and phases 𝜽𝒏 .
 From the previous example, we can
plot 𝑪𝒏 as a function of 𝝎𝒏 . This is
called the amplitude spectrum.
 We can also plot 𝜽𝒏 as a function of
𝝎𝒏 . This is called the phase spectrum.
 These two plots together are known as
the frequency spectra of 𝒈 𝒕 .
 These spectra give the
frequency domain
description of 𝒈 𝒕 .
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Slide 50
Existence of the Fourier Series
 It is important to keep in mind that the Fourier series exists subject to the
function 𝒈 𝒕 being integrable over one period.
 This means that:
න 𝒈 𝒕 𝒅𝒕 < ∞
𝑻𝟎
 This is known as the weak Dirichlet condition.
 This condition guarantees the existence of the series but not its convergence.
 Convergence is guaranteed by the strong Dirichlet condition which in addition
to the weak condition requires that 𝒈 𝒕 has a finite number of maxima and
minima in one period.
 Note: all waveforms generated in the lab satisfy the strong condition and thus
possess a convergent Fourier series.
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Slide 51
Exponential Fourier Series
 The set of exponentials 𝒆𝒋𝒏𝝎𝟎𝒕 is orthogonal over the interval 𝑻𝟎 = 𝟐𝝅Τ𝝎𝟎.
 Hence, a signal 𝒈 𝒕 can be expressed as the Fourier series:
+∞
𝒈 𝒕 = ෍ 𝑫𝒏 𝒆
𝒋𝒏𝝎𝟎 𝒕
𝟏
𝑫𝒏 =
න 𝒈 𝒕 𝒆−𝒋𝒏𝝎𝟎𝒕 𝒅𝒕
𝑻𝟎 𝑻𝟎
with
−∞
 This is simply another form of the trigonometric Fourier series.
 Note that for 𝒏 = 𝟎, 𝑫𝟎 = 𝑪𝟎 leading to:
+∞
𝒈 𝒕 = 𝑫𝟎 +
𝑫𝒏 𝒆𝒋𝒏𝝎𝟎 𝒕
෍
𝒏=−∞,𝒏≠𝟎
 This representation of the Fourier series is more convenient than the
trigonometric form and thus will be used throughout this course.
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Example 2.10
Slide 52
 Recall the signal 𝒈 𝒕 = 𝒆−𝒕Τ𝟐 for 𝟎 ≤ 𝒕 ≤ 𝝅, which we saw in example 2.7.
 We would like to derive the exponential Fourier series for it.
 We have: 𝑻𝟎 = 𝝅, 𝝎𝟎 = 𝟐𝝅ൗ𝑻𝟎 = 𝟐, and:
𝟏 𝝅 −𝒕Τ𝟐 −𝒋𝟐𝒏𝒕
𝑫𝒏 = න 𝒆
𝒆
𝒅𝒕
𝝅 𝟎
𝟏 𝝅 − 𝟏+𝒋𝟐𝒏 𝒕
= න 𝒆 𝟐
𝒅𝒕
𝝅 𝟎
𝝅
𝟏
−𝟏
𝟎. 𝟓𝟎𝟒𝟑
− 𝟐+𝒋𝟐𝒏 𝒕
=
𝒆
ቮ = 𝟏 + 𝒋𝟒𝒏
𝟏
𝝅 𝟐 + 𝒋𝟐𝒏
𝟎
 Hence,
+∞
𝟏
𝒈 𝒕 = 𝟎. 𝟓𝟎𝟒𝟑 ෍
𝒆𝒋𝟐𝒏𝒕
𝟏 + 𝒋𝟒𝒏
−∞
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Slide 53
Exponential Fourier Series
 In the exponential series, coefficients 𝑫𝒏
are complex:
𝑫𝒏 = 𝑫𝒏 𝒆𝒋𝜽𝒏
 The frequency spectra of the signal 𝒈 𝒕
can be obtained from the exponential
series by taking the magnitude and
angle of 𝑫𝒏 as functions of the
frequency 𝝎.
 Note that the spectra exist for positive
and negative frequencies.
 What is a negative frequency?
 It is difficult to imagine a negative frequency. We simply think of the negative
frequencies as an indication that an exponential exists in the series.
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Example 2.11
Slide 54
 Find the exponential Fourier
series for the square wave
𝒘 𝒕 .
 We have: 𝑻𝟎 = 𝟐𝝅, 𝝎𝟎 = 𝟐𝝅ൗ𝑻𝟎 = 𝟏, and:
𝟏 +𝝅Τ𝟐 −𝒋𝒏𝒕
𝑫𝒏 =
න
𝒆
𝒅𝒕
𝟐𝝅 −𝝅Τ𝟐
−𝟏 −𝒋𝒏𝒕 +𝝅Τ𝟐
=
𝒆
ቚ
𝒋𝟐𝒏𝝅
−𝝅Τ𝟐
−𝟏
𝟏
𝒏𝝅
=
𝒆−𝒋𝒏𝝅Τ𝟐 − 𝒆𝒋𝒏𝝅Τ𝟐 =
𝐬𝐢𝐧
𝒋𝟐𝒏𝝅
𝒏𝝅
𝟐
 Hence,
+∞
𝟏
𝒏𝝅
𝒘 𝒕 =෍
𝐬𝐢𝐧
𝒏𝝅
𝟐
−∞
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Slide 55
Parseval’s Theorem
 We saw earlier that a periodic signal is a power signal.
 Each term in its Fourier series is also a power signal.
 The power of 𝒈 𝒕 based on its compact Fourier series is:
+∞
𝑷𝒈 = 𝑪𝟐𝟎 +
𝟏
෍ 𝑪𝟐𝒏
𝟐
𝒏=𝟏
 Similarly, based on the exponential Fourier series:
+∞
+∞
𝑷𝒈 = ෍ 𝑫𝒏 𝟐 = 𝑫𝟐𝟎 + 𝟐 ෍ 𝑫𝒏 𝟐
𝒏=−∞
𝒏=𝟏
 These formulas appear different from the one seen earlier but generally state
the same principle.
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Discrete Fourier Transform
Slide 56
 In the exponential Fourier series, the coefficients 𝑫𝒏 can be calculated by
means of the discrete Fourier transform (DFT).
 Samples are taken over one period 𝑻𝟎 of 𝒈 𝒕 at an interval 𝑻𝒔 .
 The number of samples is 𝑵𝟎 = 𝑻𝟎ൗ𝑻𝒔.
 We have:
𝑵𝟎 −𝟏
𝟏
𝑫𝒏 = 𝐥𝐢𝐦
෍ 𝒈 𝒌𝑻𝒔 𝒆−𝒋𝒏𝛀𝟎 𝒌
𝑻𝒔 →𝟎 𝑵𝟎
𝒌=𝟎
 Here, 𝒈 𝒌𝑻𝒔 is the 𝒌th sample and 𝛀𝟎 = 𝝎𝟎 𝑻𝒔 .
 Note that in practice, making 𝑻𝒔 → 𝟎 is impossible. Practically, we can have:
𝑵𝟎 −𝟏
𝟏
𝑫𝒏 =
෍ 𝒈 𝒌𝑻𝒔 𝒆−𝒋𝒏𝛀𝟎𝒌
𝑵𝟎
𝒌=𝟎
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Discrete Fourier Transform
Slide 57
 The following is a Matlab script that
displays the spectra of 𝒈 𝒕 = 𝒆−𝒕Τ𝟐 over
the interval 𝟎 ≤ 𝒕 ≤ 𝝅 :
T0 = pi; N0 = 256; Ts = T0/N0; M = 10;
t = (0:Ts:Ts*(N0-1))';
g = exp(-t/2); g(1) = 0.604;
Dn = fft(g)/N0;
[Dnangle,Dnmag] =
cart2pol(real(Dn),imag(Dn));
k = (0:length(Dn)-1)';
subplot(211); stem(k,Dnmag);
axis([0 15 0 0.6]);
subplot(212); stem(k,Dnangle);
axis([0 15 -2 2]);
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Slide 58
Fourier Transform
 The Fourier series seen so far is only suitable for periodic signals.
 For aperiodic signals, we define the direct Fourier transform as:
+∞
𝑮 𝝎 =න
𝒈 𝒕 𝒆−𝒋𝝎𝒕 𝒅𝒕
−∞
 The function 𝑮 𝝎 forms the continuous envelop of the discrete coefficients 𝑫𝒏
from the exponential Fourier series.
 The inverse Fourier transform is given by:
𝟏 +∞
𝒈 𝒕 =
න 𝑮 𝝎 𝒆𝒋𝝎𝒕 𝒅𝒕
𝟐𝝅 −∞
 Symbolically, we write:
𝑮 𝝎 =ℱ 𝒈 𝒕
and
𝒈 𝒕 = ℱ −𝟏 𝑮 𝝎
 The Fourier transform pair 𝒈 𝒕 ⇔ 𝑮 𝝎 helps us understand communications.
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Slide 59
Fourier Transform
 Function 𝑮 𝝎 is the frequency domain representation of 𝒈 𝒕 .
 Since 𝑮 𝝎 is complex, it can be visualized by taking its magnitude and phase
separately:
𝑮 𝝎 = 𝑮 𝝎 𝒆𝒋𝜽𝒈 𝝎
 The function 𝑮 𝝎 is conjugately symmetric as:
𝑮 −𝝎 = 𝑮∗ 𝝎
 As a result, when the signal 𝒈 𝒕 is real, the amplitude spectrum 𝑮 𝝎 is an
even function and the phase spectrum 𝜽𝒈 𝝎 is an odd function of 𝝎.
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Example 3.1
Slide 60
 Find the Fourier transform of the signal 𝒈 𝒕 = 𝒆−𝒂𝒕 𝒖 𝒕 with 𝒂 > 𝟎.
 We have:
+∞
𝑮 𝝎 =න
𝒈 𝒕 𝒆−𝒋𝝎𝒕 𝒅𝒕 = න
−∞
+∞
=න
𝟎
+∞
𝒆−𝒂𝒕 𝒆−𝒋𝝎𝒕 𝒅𝒕
𝟎
+∞
−𝟏
−
𝒂+𝒋𝝎
𝒕
𝒆
𝒅𝒕 =
𝒆− 𝒂+𝒋𝝎 𝒕 ቤ
𝒂 + 𝒋𝝎
𝟎
𝟏
=
𝒂 + 𝒋𝝎
 Taking the magnitude and phase yields:
𝑮 𝝎
=
𝟏
𝒂𝟐 + 𝝎 𝟐
𝝎
−𝟏
𝜽𝒈 𝝎 = − 𝐭𝐚𝐧
𝒂
 As expected, 𝑮 𝝎 is
even and 𝜽𝒈 𝝎 is odd.
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Characteristics of the Fourier Transform
Slide 61
 The Fourier transform exists subject to the same Dirichlet conditions seen earlier
for the Fourier series.
 The first Dirichlet condition is sufficient but not necessary. For instance,
𝐬𝐢𝐧 𝒂𝒕
𝒈 𝒕 =
𝒕
has a Fourier transform even though the condition is not satisfied.
 The Fourier transform is linear meaning that if:
𝒈𝟏 𝒕 ⇔ 𝑮𝟏 𝝎 and 𝒈𝟐 𝒕 ⇔ 𝑮𝟐 𝝎
then,
𝒂𝟏 𝒈𝟏 𝒕 + 𝒂𝟐 𝒈𝟐 𝒕 ⇔ 𝒂𝟏 𝑮𝟏 𝝎 + 𝒂𝟐 𝑮𝟐 𝝎
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Slide 62
Common Fourier Pairs
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Slide 63
Properties of the Fourier Transform
 Given 𝒈 𝒕 ⇔ 𝑮 𝝎 , the symmetry property states that:
𝑮 𝒕 ⇔ 𝟐𝝅𝒈 −𝝎
 The scaling property states that:
𝟏
𝝎
𝒈 𝒂𝒕 ⇔
𝑮
𝒂
𝒂
 The scaling property means that
compression in time leads to
expansion in frequency and
vice versa.
 This leads to reciprocity of signal
duration and its bandwidth.
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Slide 64
Properties of the Fourier Transform
 The Time shifting property states that:
𝒈 𝒕 − 𝒕𝟎 ⇔ 𝑮 𝝎 𝒆−𝒋𝝎𝒕𝟎
 Note how the amplitude spectrum is unchanged as 𝒆−𝒋𝝎𝒕𝟎 = 𝟏, whereas the
phase spectrum changes by −𝝎𝒕𝟎 .
 For instance, time shifting 𝒈 𝒕 = 𝐜𝐨𝐬 𝝎𝒕 yields:
𝒈 𝒕 = 𝐜𝐨𝐬 𝝎 𝒕 − 𝒕𝟎 = 𝐜𝐨𝐬 𝝎𝒕 − 𝝎𝒕𝟎
 The frequency shifting property states that:
𝒈 𝒕 𝒆𝒋𝝎𝟎𝒕 ⇔ 𝑮 𝝎 − 𝝎𝟎
 This means that shifting the spectrum by 𝝎𝟎 is equivalent to multiplying 𝒈 𝒕 by
the exponential 𝒆𝒋𝝎𝟎 𝒕 .
 This property will be useful when we talk about amplitude modulation (AM).
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Slide 65
Properties of the Fourier Transform
 The convolution of functions 𝒈 𝒕 and 𝒘 𝒕 is given by:
+∞
𝒈 𝒕 ∗𝒘 𝒕 =න
𝒈 𝝉 𝒘 𝒕 − 𝝉 𝒅𝝉
−∞
 The time convolution property states that:
𝒈𝟏 𝒕 ∗ 𝒈𝟐 𝒕 ⇔ 𝑮𝟏 𝝎 𝑮𝟐 𝝎
 Similarly, the frequency convolution property states that:
𝟏
𝒈𝟏 𝒕 𝒈𝟐 𝒕 ⇔
𝑮𝟏 𝝎 ∗ 𝑮𝟐 𝝎
𝟐𝝅
 The time differentiation property states that:
𝒅𝒈
⇔ 𝒋𝝎𝑮 𝝎
𝒅𝒕
 The time integration property states that:
𝒕
𝟏
න 𝒈 𝝉 𝒅𝝉 =
𝑮 𝝎 + 𝝅𝑮 𝟎 𝜹 𝝎
𝒋𝝎
−∞
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Slide 66
Summary of Fourier Transform Operations
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Slide 67
End of Topic 2
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