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Transcript
COMM 1208 Unit 2 Signals Signals 1. 2. 3. 4. 5. 6. Introduction to Signals ..................................................................................................... 2 Frequency ............................................................................................................................ 3 Definition of a Sinusoidal Signal .................................................................................... 4 Cosine Wave ........................................................................................................................ 5 Time Domain and Frequency Domain Representations ............................................ 6 Filtering ............................................................................................................................... 8 6.1 Low pass filter ................................................................................................................ 8 6.2 High pass filter............................................................................................................... 9 6.3 Band pass filter ............................................................................................................ 10 6.4 Band stop filter ............................................................................................................ 10 7. Power Levels ..................................................................................................................... 10 8. Fourier Analysis ............................................................................................................... 11 Page 1 of 14 1 COMM 1208 Unit 2 Signals 1. Introduction to Signals There are various forms of electrical signals that are encountered in communications systems. Examples include: Continuous analogue Discrete digital Square wave Sinusoidal Noise Signal Pulsed Signal The continuous analogue signal varies continuously between a minimum level and a maximum level. It could be representative of a speech or audio signal for example. The discrete digital signal assumes a finite number of logic levels. A 2 level signal is shown in the example above. Typically a high voltage corresponds to a logic 1 level and a low voltage as a logic 0 level. The digital signal could be data from a computer or it might be a digital representation of an analogue signal that has been passed through an analogue to digital converter. The square wave is a 2 level signal that has a 1 0 1 0 1 0 ….. sequence. It could be used as a test signal or it could represent a data clock. A sinusoidal signal has a specific frequency and amplitude. It is often used for testing analogue systems. A noise signal is random in nature. This signal is found as an interference. It tends to derate the performance of a system. A pulsed signal is similar to a square wave except that the duration of the high and low levels are not equal. This type of signal can be used to produce sampling of analogue signals. Page 2 of 14 2 COMM 1208 Unit 2 Signals 2. Frequency Frequency = For a periodic function, the number of cycles or events per unit time. Hertz = The SI unit of frequency, equal to one cycle per second. Frequency = 1 Time 1 Second = 1 Hertz 1 Hertz 1,000 Hertz = 1 kHertz 1,000,000 Hertz = 1,000 kHertz = 1MHertz 1,000,000,000 Hertz = 1,000,000 kHertz = 1,000 MHertz = 1 GHertz 1,000,000,000,000 Hertz = 1,000,000,000 kHertz = 1,000,000 MHertz = 1,000 GHertz = 1 THertz 1 kHertz = 1 kilo Hertz 1MHertz = 1 Mega Hertz 1 GHertz = 1 Giga Hertz 1 THertz = 1 Tera Hertz Page 3 of 14 3 COMM 1208 Unit 2 Signals 3. Definition of a Sinusoidal Signal A continuous analogue sinusoidal signal is defined by: s(t ) A sin(2 f1t ) A sin(1t ) The amplitude of this sinusoidal signal or tone is A and the frequency is f1. The frequency is specified in cycles per second or Hertz. The frequency can also be specified in radians per second or rads. This is the term 1 where: 1 2 f1t 1 kHz Tone of amplitude 10 V 10 8 6 Amplitude (V) 4 2 0 -2 -4 -6 -8 -10 0 0.5 1 Time ms 1.5 2 This plots is an example of a single tone of amplitude A=10, and of frequency f1=1 kHz. Note that the period of this signal is 1 ms. Page 4 of 14 4 COMM 1208 Unit 2 Signals 4. Cosine Wave A cosine wave is a signal waveform with a shape identical to that of a sine wave, except each point on the cosine wave occurs exactly 1/4 cycle earlier than the corresponding point on the sine wave. A cosine wave and its corresponding sine wave have the same frequency, but the cosine wave leads the sine wave by 90 degrees of phase. The cosine wave is known as an even function because the curve is symmetrical about the vertical axis. A sine wave is an odd function because the curve is skew-symmetrical about the vertical axis. S(t) = A Cos(2πf1t) A more general expression is S(t) = A Cos(2πf1t - τ) In terms of a phase angle this can be written S(t) = A Cos(2πf1t + φ) Where φ = -2π τ/T1 A numerically negative value of φ results in a lagging phase angle, and a positive value a leading phase angle. For a Sine wave τ = T1/4 Page 5 of 14 5 COMM 1208 Unit 2 Signals 5. Time Domain and Frequency Domain Representations The previous representation of the analogue tone is said to be a time domain representation. It corresponds to the plot of the amplitude of the signal as a function of time. This plot can be obtained by computer simulation, or it could also be obtained by using a cathode ray oscilloscope (CRO) to display a real sinusoidal signal. Another way to view this signal is in the frequency domain. This plots the amplitude of each frequency component of a signal against a horizontal axis of frequency rather than time. The advantage of the frequency domain representation is that it can display clearly the various frequency components that exist in a signal. The frequency domain representation or spectrum of the single tone is shown in the diagram below. AV Frequency (Hz) f1 Page 6 of 14 6 COMM 1208 Unit 2 Signals Consider the following analogue signal: s(t ) 3 sin( 2f1t ) sin( 2 (3 f1 )t ) This signal consists of the addition of 2 frequency components. The time domain and frequency domain representations of this signal are shown below. 1 kHz Tone + of 3 kHz Tone of amplitude 3 V and 1 V resp 3 2 Amplitude (V) 1 0 -1 -2 -3 0 0.5 1 Time ms 1.5 2 3V 1V f1 3f1 Page 7 of 14 Frequency (Hz) 7 COMM 1208 Unit 2 Signals 6. Filtering The process of filtering is to allow various frequencies within a signal to pass through the filter and to block others. An example of a filter is a camera filter that allows certain light frequencies only to pass through the camera lens. Typical filter responses are: Low-pass High-pass Band-pass Band-stop 6.1 Low pass filter A low-pass filter allows low frequencies through, but removes or attenuates sufficiently medium and high frequencies. The characteristics of a filter can be shown in an amplitude response plot of the filter. This is a plot of the amplitude that will occur at the output of the filter for a range of input frequencies. This is called the filter frequency response. The frequency response is obtained by applying a sinusoidal signal to the input of the filter and plotting the output level for various frequencies. Sinusoidal Generator Variable Filter Sinusoidal Output Digital Voltmeter Measuring the frequency response of a filter Amplitude Frequency Low-pass filter frequency response Page 8 of 14 8 COMM 1208 Unit 2 Signals Sometimes a filter is assumed to be ideal for the sake of simplifying the analysis of a system. The ideal low-pass filter for example, passes all frequencies up to its cut-off frequency and completely rejects all frequencies above this frequency. Amplitude Cut-off frequency Frequency Ideal Low-pass filter frequency response Composite Signal s(t ) 3 sin( 2f1t ) sin( 2 (3 f1 )t ) If the composite signal previously analysed consisting of the 2 frequency components of 1 kHz and 3 kHz is connected to the input of an ideal low-pass filter of cut-off frequency 2 kHz or 2f1, then the output signal will be a single 1 kHz frequency component whose amplitude is 3 V. The output would therefore be: s(t ) 3sin(2 f1t ) 6.2 High pass filter Ideal filter response Amplitude Stops frequencies below the cutoff frequency Allows frequencies above the cutoff frequency to pass Frequency Cutoff frequency Ideal High-pass filter frequency response Page 9 of 14 9 COMM 1208 Unit 2 Signals 6.3 Band pass filter Stops frequencies either side of fstart and fstop Amplitude Allows freqs above fstart and below fstop to pass Ideal filter response frequency fstart fstop Ideal Band-pass filter frequency response 6.4 Band stop filter Allows freqs below fstart and above fstop to pass Amplitude Stops freqs in between Ideal filter response frequency fstart fstop Ideal Band-Stop filter frequency response 7. Power Levels The peak power level of a sinusoidal signal is obtained by dividing the square of the peak amplitude by the resistance. For example, if the above tone is connected to a 600Ω load resistance the peak power level across the load is V02 32 9 P0 ( peak ) 15mW R 600 600 The RMS power level (type of average) is obtained by squaring the root-mean squared voltage (rms) instead of the peak level. 2 V0 2 32 9 2 V0 P0 (average) 7.5mW R 2 R 2 600 1200 Page 10 of 14 10 COMM 1208 Unit 2 Signals 8. Fourier Analysis A mathematical process known as Fourier Analysis shows that any waveform can be treated as if it is made up of a number of sine waves of different frequencies, amplitudes and phase relationships. For example a square wave which has amplitude Vm and frequency f is made up of a series of sine waves with frequencies, f, 3f, 5f etc. and can be described by the following equation: Vm t V (t ) 4 Vm 1 1 1 sin( 2 f t ) sin( 2 3 f t ) sin( 2 5 f ot ) ......... o o 1 3 5 Where …indicates that the sum goes on for ever. This can also be written as V (t ) 4 Vm 1 sin( 2kfot ) k odd , k 1 k A closer approximation to a square wave is obtained by adding in more and more terms and can be seen in the diagram. Page 11 of 14 11 COMM 1208 Unit 2 Signals -0.5 41 36 31 21 26 41 36 31 26 16 11 21 36 41 36 41 31 26 21 1 41 7 terms to sin13x/13 0 -0.2 -0.4 -0.6 -0.8 -1 6 6 terms to sin11x / 11 1 0.8 0.6 0.4 0.2 36 31 26 21 16 6 1 11 5 terms to sin 9x/9 1 0.8 0.6 0.4 0.2 16 -1 11 -0.5 sin x+sin3x/3+sin5x/5+sin 7x/7 6 41 36 31 26 21 16 11 6 1 0 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 1 sin x + 1/3 sin 3x + 1/5 sin 5x 0.5 8 terms to sin15x/15 1 0.8 31 26 21 16 11 6 -0.2 -0.4 -0.6 1 41 36 31 26 21 16 11 6 0.6 0.4 0.2 0 1 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 11 -0.6 -0.8 -1 -1 0 -0.2 -0.4 -0.6 -0.8 -1 6 41 36 31 26 21 16 11 6 1 0 0.4 0.2 0 -0.2 -0.4 1 0.5 1 sin x + 1/3 sin3x 1 0.8 0.6 16 sin x 1 -0.8 -1 Page 12 of 14 12 COMM 1208 Unit 2 Signals A triangular wave can be described by V (t ) 8 Vm 1 1 1 sin( 2 f t ) sin( 2 3 f t ) sin( 2 5 f t ) ......... o o o 2 1 9 25 Or V (t ) 8 Vm 1 sin( 2kfo t ) 2 k odd , k 1 2 k Vm t A sawtooth wave is described by V(t) = 2 * Vm/ (Sin t - 1/2 Sin 2t + 1/3 Sin 3t - 1/4 Sin 4t + 1/5 Sin 5t + ….) V (t ) 2 Vm 1 1 1 1 sin( 2f ot ) sin( 2 2 f ot ) sin( 2 3 f ot ) sin( 2 4 f ot )......... 1 2 3 4 The diagrams above show the time representation of a signal, or it shows the time domain of the signal i.e. how the voltage of the signal varies with time. Alternatively we can produce a frequency diagram of the signal, also called a diagram in the frequency domain, or a frequency spectrum of the signal. The frequency spectrum of a square wave is shown below. For a simple signal, such as a square wave both the time and frequency domain diagrams give all of the information about the signal. For a more complex waveform, such as music, speech or a digitised signal the frequency domain diagram will give only a very rough impression of the signal in the time domain. A square wave has an amplitude of 10 volts and a frequency of 1 kHz Vm t In the time domain Page 13 of 14 13 COMM 1208 Unit 2 Signals V (t ) 4 10 1 1 1 sin( 2 1000t ) sin( 2 3000t ) sin( 2 5000t ) ......... 1 3 5 V (t ) 12.73sin( 21000t ) 4.24 sin( 2 3000t ) 2.54 sin( 2 5000t ) ......... Vm in Volts 12.73 V 4.24 V 2.54 V 1.81 V f in kHz In the frequency domain Page 14 of 14 14