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. Communication theory & systems (EE372) | Dr. Samir Bendoukha Home 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. Communication theory & systems (EE372) | Dr. Samir Bendoukha Home 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 Communication theory & systems (EE372) | Dr. Samir Bendoukha Home 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. Communication theory & systems (EE372) | Dr. Samir Bendoukha Home 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. Communication theory & systems (EE372) | Dr. Samir Bendoukha Home 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. Communication theory & systems (EE372) | Dr. Samir Bendoukha Home 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 . Communication theory & systems (EE372) | Dr. Samir Bendoukha Home 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. Communication theory & systems (EE372) | Dr. Samir Bendoukha Home 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. Communication theory & systems (EE372) | Dr. Samir Bendoukha Home 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. Communication theory & systems (EE372) | Dr. Samir Bendoukha Home 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. Communication theory & systems (EE372) | Dr. Samir Bendoukha Home 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 . Communication theory & systems (EE372) | Dr. Samir Bendoukha Home 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. Communication theory & systems (EE372) | Dr. Samir Bendoukha Home 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. Communication theory & systems (EE372) | Dr. Samir Bendoukha Home 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 = −∞ Communication theory & systems (EE372) | 𝒅𝒖 𝒅𝒕 Dr. Samir Bendoukha Home 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. Communication theory & systems (EE372) | Dr. Samir Bendoukha Home 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 Communication theory & systems (EE372) | Dr. Samir Bendoukha Home 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 Communication theory & systems (EE372) | Dr. Samir Bendoukha Home 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. Communication theory & systems (EE372) | Dr. Samir Bendoukha Home 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 Communication theory & systems (EE372) | Dr. Samir Bendoukha Home 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 Communication theory & systems (EE372) 𝜽𝒏 = 𝐭𝐚𝐧 | −𝟏 −𝒃𝒏 𝒂𝒏 Dr. Samir Bendoukha Home 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 Communication theory & systems (EE372) | Dr. Samir Bendoukha Home 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 Communication theory & systems (EE372) | 𝟎≤𝒕≤𝝅 Dr. Samir Bendoukha Home 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 . Communication theory & systems (EE372) | Dr. Samir Bendoukha Home 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. Communication theory & systems (EE372) | Dr. Samir Bendoukha Home 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. Communication theory & systems (EE372) | Dr. Samir Bendoukha Home 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 −∞ Communication theory & systems (EE372) | Dr. Samir Bendoukha Home 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. Communication theory & systems (EE372) | Dr. Samir Bendoukha Home 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 −∞ Communication theory & systems (EE372) | Dr. Samir Bendoukha Home 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. Communication theory & systems (EE372) | Dr. Samir Bendoukha Home 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 Communication theory & systems (EE372) | Dr. Samir Bendoukha Home 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]); Communication theory & systems (EE372) | Dr. Samir Bendoukha Home 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. Communication theory & systems (EE372) | Dr. Samir Bendoukha Home 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. Communication theory & systems (EE372) | Dr. Samir Bendoukha Home 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. Communication theory & systems (EE372) | Dr. Samir Bendoukha Home 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 Communication theory & systems (EE372) | Dr. Samir Bendoukha Home Slide 62 Common Fourier Pairs Communication theory & systems (EE372) | Dr. Samir Bendoukha Home 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. Communication theory & systems (EE372) | Dr. Samir Bendoukha Home 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). Communication theory & systems (EE372) | Dr. Samir Bendoukha Home 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 −∞ Communication theory & systems (EE372) | Dr. Samir Bendoukha Home Slide 66 Summary of Fourier Transform Operations Communication theory & systems (EE372) | Dr. Samir Bendoukha Home Slide 67 End of Topic 2 Any Questions? Communication theory & systems (EE372) | Dr. Samir Bendoukha Home
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