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**Sine wave Fourier series Fourier transform**

Chapter 8 Sine wave Fourier series Fourier transform 09/16/2010 © 2010 NTUST

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Wave A wave is a disturbance. Unlike water waves, electrical waves cannot be seen directly but they have similar characteristics. All periodic waves can be constructed from sine waves, which is why sine waves are fundamental.

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Sine Waves The sinusoidal waveform (sine wave) is the fundamental alternating current (ac) and alternating voltage waveform. Electrical sine waves are named from the mathematical function with the same shape.

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Period of a Sine Wave

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Sine Waves Sine waves are characterized by the amplitude and period. The amplitude is the maximum value of a voltage or current; the period is the time interval for one complete cycle. Example A The amplitude (A) of this sine wave is 20 V T The period is 50.0 ms

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Sine Waves The period of a sine wave can be measured between any two corresponding points on the waveform. T T T T A T T By contrast, the amplitude of a sine wave is only measured from the center to the maximum point.

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Frequency Frequency ( f ) is the number of cycles that a sine wave completes in one second. Frequency is measured in hertz (Hz). Example If 3 cycles of a wave occur in one second, the frequency is 3.0 Hz 1.0 s

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**Frequency of a Sine Wave**

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Period and Frequency The period and frequency are reciprocals of each other. and Thus, if you know one, you can easily find the other. (The 1/x key on your calculator is handy for converting between f and T.) Example If the period is 50 ms, the frequency is 0.02 MHz = 20 kHz.

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**Generation of a Sine Wave**

Sinusoidal voltages are produced by ac generators and electronic oscillators. When a conductor rotates in a constant magnetic field, a sinusoidal wave is generated. D C B A When the loop is moving perpendicular to the lines of flux, the maximum voltage is induced. When the conductor is moving parallel with the lines of flux, no voltage is induced.

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**AC Generator (Alternator)**

Generators convert rotational energy to electrical energy. A stationary field alternator with a rotating armature is shown. The armature has an induced voltage, which is connected through slip rings and brushes to a load. The armature loops are wound on a magnetic core (not shown for simplicity). Small alternators may use a permanent magnet as shown here; other use field coils to produce the magnetic flux.

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**AC Generator (Alternator)**

By increasing the number of poles, the number of cycles per revolution is increased. A four-pole generator will produce two complete cycles in each revolution.

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**Function Generator Readout Typical controls: Function selection**

Frequency Range Adjust Outputs Output level (amplitude) Duty cycle DC offset CMOS output

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**Sine Wave Voltage and Current**

There are several ways to specify the voltage of a sinusoidal voltage waveform. The amplitude of a sine wave is also called the peak value, abbreviated as VP for a voltage waveform. Example VP The peak voltage of this waveform is 20 V.

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**Sine Wave Voltage and Current**

The voltage of a sine wave can also be specified as either the peak-to-peak or the rms value. The peak-to-peak is twice the peak value. The rms value is times the peak value. Example The peak-to-peak voltage is Vrms 40 V. VPP The rms voltage is 14.1 V.

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**Sine Wave Voltage and Current**

For some purposes, the average value (actually the half-wave average) is used to specify the voltage or current. By definition, the average value is as times the peak value. Example The average value for the sinusoidal voltage is Vavg 12.7 V.

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**Sine Wave Voltage and Current**

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**Sine Wave Voltage and Current**

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**Sine Wave Voltage and Current**

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**Sine Wave Voltage and Current**

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**Sine Wave Voltage and Current**

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**Sine Wave Voltage and Current**

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Angular Measurement Angular measurements can be made in degrees (o) or radians. The radian (rad) is the angle that is formed when the arc is equal to the radius of a circle. There are 360o or 2p radians in one complete revolution.

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Angular Measurement

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Angular Measurement

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Angular Measurement Because there are 2p radians in one complete revolution and 360o in a revolution, the conversion between radians and degrees is easy to write. To find the number of radians, given the number of degrees: To find the number of degrees, given the radians:

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Sine Wave Equation

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Sine Wave Equation Instantaneous values of a wave are shown as v or i. The equation for the instantaneous voltage (v) of a sine wave is where Vp = Peak voltage q = Angle in rad or degrees Example If the peak voltage is 25 V, the instantaneous voltage at 50 degrees is 19.2 V

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Sine Wave Equation

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Examples

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Examples

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Sine Wave Equation A plot of the example in the previous slide (peak at 25 V) is shown. The instantaneous voltage at 50o is 19.2 V as previously calculated.

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Examples

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Examples

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Phasor The sine wave can be represented as the projection of a vector rotating at a constant rate. This rotating vector is called a phasor. Phasors are useful for showing the phase relationships in ac circuits.

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Phase Shift The phase of a sine wave is an angular measurement that specifies the position of a sine wave relative to a reference. To show that a sine wave is shifted to the left or right of this reference, a term is added to the equation given previously. where f = Phase shift

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**Phase Shift …and the equation has a negative phase shift**

Example of a wave that lags the reference …and the equation has a negative phase shift v = 30 V sin (q - 45o) Notice that a lagging sine wave is below the axis at 0o

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**Phase Shift Notice that a leading sine wave is above the axis at 0o**

Example of a wave that leads the reference Notice that a leading sine wave is above the axis at 0o v = 30 V sin (q + 45o) …and the equation has a positive phase shift

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**Power in Resistive AC Circuits**

The power relationships developed for dc circuits apply to ac circuits except you must use rms values when calculating power. The general power formulas are:

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**Power in Resistive AC Circuits**

Assume a sine wave with a peak value of 40 V is applied to a 100 W resistive load. What power is dissipated? Example Solution Vrms = x Vp = x 40 V = V 8 W

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Instantaneous Value

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**Superimposed DC and AC Voltage**

Frequently dc and ac voltages are together in a waveform. They can be added algebraically, to produce a composite waveform of an ac voltage “riding” on a dc level.

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**Superimposed DC and AC Voltage**

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Examples

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Examples

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Examples

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Examples

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Pulse Definitions Ideal pulses

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**Pulse Definitions Non-ideal pulses**

Notice that rise and fall times are measured between the 10% and 90% levels whereas pulse width is measured at the 50% level.

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Repetitive Pulses

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Examples

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Examples

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**Triangular and Sawtooth Wave**

Triangular and sawtooth waveforms are formed by voltage or current ramps (linear increase/decrease) Triangular waveforms have positive-going and negative-going ramps of equal duration. The sawtooth waveform consists of two ramps, one of much longer duration than the other.

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Harmonics All repetitive non-sinusoidal waveforms are composed of a fundamental frequency (repetition rate of the waveform) and harmonic frequencies. Odd harmonics are frequencies that are odd multiples of the fundamental frequency. Even harmonics are frequencies that are even multiples of the fundamental frequency.

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Harmonics A square wave is composed only of the fundamental frequency and odd harmonics (of the proper amplitude).

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Oscilloscope The oscilloscope is divided into four main sections.

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**© Copyright 2007 Prentice-Hall**

Oscilloscope © Copyright 2007 Prentice-Hall

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Oscilloscopes Display Trigger Horizontal Vertical

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Selected Key Terms A type of waveform that follows a cyclic sinusoidal pattern defined by the formula y = A sin q. Sine wave Alternating current Period (T) Frequency (f) Hertz Current that reverses direction in response to a change in source voltage polarity. The time interval for one complete cycle of a periodic waveform. A measure of the rate of change of a periodic function; the number of cycles completed in 1 s. The unit of frequency. One hertz equals one cycle per second.

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**Selected Key Terms Instantaneous value**

Peak value Peak-to-peak value rms value The voltage or current value of a waveform at a given instant in time. The voltage or current value of a waveform at its maximum positive or negative points. The voltage or current value of a waveform measured from its minimum to its maximum points. The value of a sinusoidal voltage that indicates its heating effect, also known as effective value. It is equal to times the peak value. rms stands for root mean square.

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Selected Key Terms Radian Phase Amplitude Pulse Harmonics A unit of angular measurement. There are 2p radians in one complete 360o revolution. The relative angular displacement of a time-varying waveform in terms of its occurrence with respect to a reference. The maximum value of a voltage or current. A type of waveform that consists of two equal and opposite steps in voltage or current separated by a time interval. The frequencies contained in a composite waveform, which are integer multiples of the pulse repetition frequency.

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Quiz 1. In North America, the frequency of ac utility voltage is 60 Hz. The period is a. 8.3 ms b ms c. 60 ms d. 60 s

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**Quiz 2. The amplitude of a sine wave is measured**

a. at the maximum point b. between the minimum and maximum points c. at the midpoint d. anywhere on the wave

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Quiz 3. An example of an equation for a waveform that lags the reference is a. v = -40 V sin (q) b. v = 100 V sin (q + 35o) c. v = 5.0 V sin (q - 27o) d. v = 27 V

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**Quiz 4. In the equation v = Vp sin q , the letter v stands for the**

a. peak value b. average value c. rms value d. instantaneous value

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Quiz 5. The time base of an oscilloscope is determined by the setting of the a. vertical controls b. horizontal controls c. trigger controls d. none of the above

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**Quiz 6. A sawtooth waveform has**

a. equal positive and negative going ramps b. two ramps - one much longer than the other c. two equal pulses d. two unequal pulses

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Quiz 7. The number of radians in 90o are a. p/2 b. p c. 2p/3 d. 2p

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Quiz 8. For the waveform shown, the same power would be delivered to a load with a dc voltage of a V b V c V d V

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**Quiz 9. A square wave consists of a. the fundamental and odd harmonics**

b. the fundamental and even harmonics c. the fundamental and all harmonics d. only the fundamental

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Quiz 10. A control on the oscilloscope that is used to set the desired number of cycles of a wave on the display is a. volts per division control b. time per division control c. trigger level control d. horizontal position control

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Quiz Answers: 1. b 2. a 3. c 4. d 5. b 6. b 7. a 8. c 9. a 10. b

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Fourier Series

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**Jean Baptiste Joseph Fourier (French)(1763~1830)**

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**Fourier Series 任一週期(periodic)函數可以分解成許多不同振幅(amplitude)，不同頻率(frequency)的**

正弦(sinusoidal)諧波(harmonic) 與 餘弦(cosinusoidal)諧波(harmonic) 的合成(composition)

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Fourier Series 傅立葉級數(Fourier Series)的基本觀念即是以弦波函數來組成信號空間，每個週期函數都可利用弦波函數來組成。 一個信號x(t) 可以表為傅立葉級數如下：

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Fourier Series A function f(x) can be expressed as a series of sines and cosines: where:

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方形波

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Adding Harmonics

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三種諧波(harmonic ) 諧波是指所有基頻波的倍頻波

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三個諧波的合成

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Square Wave Any periodic function can be expressed as the sum of a series of sines and cosines (of varying amplitudes)

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頻譜比較

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Sawtooth Wave

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Fourier Series 尤拉公式: eiφ = cosφ + isinφ 其概念與複數平面之極式相通

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Fourier Series 以複數型式表示傅立葉級數，將更為簡潔

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**Discrete Fourier Transform (DFT)**

但由於離散傅立葉所做的計算量過於龐大，當處理大量的資料時，需要快速計算的演算法。

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**Discrete Fourier Transform (DFT)**

以數位方式對連續信號取樣，週期時間T之內，可取樣N 個取樣點的數位信號 DFT 可表為 式中m為頻域上的第m個刻度，n 為時域上的第n 個刻度 X(m) 為頻域上第m個刻度向量，x(n) 為時域上第n 個刻度純量

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**Discrete Fourier Transform**

Forward DFT: Inverse DFT: The complex numbers f0 … fN are transformed into complex numbers F0 … Fn The complex numbers F0 … Fn are transformed into complex numbers f0 … fN

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DFT Example Interpreting a DFT can be slightly difficult, because the DFT of real data includes complex numbers. Basically: The magnitude of the complex number for a DFT component is the power at that frequency. The phase θ of the waveform can be determined from the relative values of the real and imaginary coefficents. Also both positive and “negative” frequencies show up.

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DFT Example

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DFT Examples

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DFT Examples

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**Fast Fourier Transform**

Discrete Fourier Transform would normally require O(n2) time to process for n samples: Don’t usually calculate it this way in practice. Fast Fourier Transform takes O(n log(n)) time. Most common algorithm is the Cooley-Tukey Algorithm.

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**Fast Fourier Transform**

FFT (Fast Fourier Transform)，大幅提高頻譜的計算速度 FFT使用條件： 信號必須是週期性的。 取樣週期必須為信號週期的整數倍。 取樣速率(Sampling rate)必須高於信號最高頻率的2 倍以上。 取樣點數N 必須為2k個資料。

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快速傅利葉轉換原理 A complex nth root of unity is a complex number z such that zn = 1. n = e 2 i / n = principal n th root of unity. e i t = cos t + i sin t. i2 = -1. There are exactly n roots of unity: nk, k = 0, 1, , n-1. n2= n/2 nn+k= nk 2 = i 3 1 4 = -1 0 = 1 7 5 forms a group under multiplication (similar to additive group Z_n modulo n) 6 = -i

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**Fourier Cosine Transform**

Any function can be split into even and odd parts: Then the Fourier Transform can be re-expressed as:

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**Discrete Cosine Transform (DCT)**

When the input data contains only real numbers from an even function, the sin component of the DFT is 0, and theDFT becomes a Discrete Cosine Transform (DCT) There are 8 variants however, of which 4 are common.

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**DCT Types DCT Type II Used in JPEG, repeated for a 2-D transform.**

Most common DCT.

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**DCT Types DCT Type IV Used in MP3.**

In MP3, the data is overlapped so that half the data from one sample set is reused in the next. Known as Modified DCT or MDCT This reduces boundary effects.

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**Why do we use DCT for Multimedia?**

For audio: Human ear has different dynamic range for different frequencies. Transform to from time domain to frequency domain, and quantize different frequencies differently. For images and video: Human eye is less sensitive to fine detail. Transform from spacial domain to frequency domain, and quantize high frequencies more coarsely (or not at all) Has the effect of slightly blurring the image - may not be perceptable if done right.

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Why use DCT/DFT? Some tasks are much easier to handle in the frequency domain that in the time domain. Eg: graphic equalizer. We want to boost the bass: Transform to frequency domain. Increase the magnitude of low frequency components. Transform back to time domain.

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Fourier Transform Fourier Series can be generalized to complex numbers, and further generalized to derive the Fourier Transform. Forward Fourier Transform: Inverse Fourier Transform: Note:

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Fourier Transform Fourier Transform maps a time series (eg audio samples) into the series of frequencies (their amplitudes and phases) that composed the time series. Inverse Fourier Transform maps the series of frequencies (their amplitudes and phases) back into the corresponding time series. The two functions are inverses of each other.

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**Basic Properties Linearity: h(x) = aƒ(x) + bg(x) **

. Basic Properties Linearity: h(x) = aƒ(x) + bg(x) Translation: h(x) = ƒ(x − x0) Modulation: h(x) = e2πixξ0ƒ(x) Scaling: h(x) = ƒ(ax) Conjugation: Duality: Convolution:

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**Derivative Properties**

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**Basic Properties Linearity: h(x) = aƒ(x) + bg(x) **

. Basic Properties Linearity: h(x) = aƒ(x) + bg(x) Translation: h(x) = ƒ(x − x0) Modulation: h(x) = e2πixξ0ƒ(x) Scaling: h(x) = ƒ(ax) Conjugation: Duality: Convolution:

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