4Summary of our radio waveforms Audio Frequency (AF)Beat Frequency Oscillator (BFO)Intermediate Frequency stage (IF)Local Oscillator (LO)Radio Frequency stage (RF).
5The ideal mixer (A reminder) An ideal mixer multiplies rather than adds waveforms.In a moment we will look at the electronics of mixers.If you feed two sine waves at frequencies F and G into a multiplier you just get sine waves at frequencies F+G and F-G and no harmonics.Lets remind ourselves what these waveforms are like before we look in more detail at real mixers.
8Some mathsLast time we noted that the output waveforms were 90 degree phase shifted sine waves of half the amplitude.For many purposes this makes no difference. However we will look at this in more detail later in the talk (but avoiding maths).Sin(f)* Sin(g) = Cos(f-g)/2 – Cos(f+g)/2
21Advantages of the double balanced mixer Almost linear on each inputGreat carrier and input rejectionLow drive signals needed.Low harmonic distortion on both inputsWell suit to IC manufactureNo transformersCheap (due to IC process)
24OrthogonalityTwo things are orthogonal if changing one doesn’t change the other. In geometry this is a right angle.For example: Latitude, Longitude and Altitude over sea are orthogonal. Over land they are not.Sine waves of different frequencies are Orthogonal.Most other waveforms are not orthogonal.
25Orthogonality Example If you feed sine waves at frequencies F and G into a mixer you get sine waves at frequencies F+G and F-G.If F=G then you get 2F and DC outSo if you take the DC average of the output you will get zero unless F=G. (Only true for orthogonal waveforms such as sine waves)So if we use an accurate signal generator for G then the DC value is a measure of the harmonic of F at G
26The spectrum analyserIf we vary the frequency of our signal generator G into our mixer then we can measure the strength of the signal F at a range of frequencies. (Just like tuning a radio)If the signal F that we are measuring is not a pure sine wave then as we tune the generator we will only measure the sine wave component of the signal F at the frequency of our generator G.So by sweeping G we can measure the spectrum of F
27The Fourier transformPreviously we said that when you mix F and G and F=G you will get a DC average. This is only true if F and G are in phase. If F and G are antiphase you get a negative DC value.However if F and G are 90 degrees apart you will get zero. So you can measure the phase of F by measuring at both 0 and 90 degrees (I and Q).Note that sine and cosine waves at the same frequency are orthogonal.
28The Fourier transform A Fourier transform is like a spectrum analyser. Multiply the original waveform by sine waves of each harmonic in turn and take the DC averages. These give you the sine wave harmonics.Now do the same thing with cosine waves, This gives you the cosine wave harmonics. (90 degrees shifted)We will see that for a Square wave you get the 1/3, 1/5 1/7 ratios (odd harmonics) we used in the signals talk.
31Mixer output for the fundamental Note the strong positive DC average
32Mixer output for the 2nd harmonic Note the average is zero (even harmonic)
33Mixer output for the 3nd harmonic Note the 4 positive peaks 2 negative. Average is 2/6. This is 1/3 of the fundamental signal
34Mixer output for the 4th harmonic Note the average is zero (even harmonic)
35Mixer output for 5nd harmonic Note the 6 positive peaks 4 negative. Average is 2/10. This is 1/5 of the fundamental signal
36Mixer output for the 6th harmonic Note the average is zero (even harmonic)
37Mixer output for 7nd harmonic Note the 8 positive peaks 6 negative. Average is 2/14. This is 1/7 of the fundamental signal
38But what about the cosine components? So far we have only looked at the sine wave (in phase) components. We should check if there are any Cosine (90 degree phase shifted) components.Note the Cosine is symmetric about the centre
39Mixer output for the Fundamental Cosine Note the average is zero (anti-symmetric about centre)
40Mixer output for the 2nd Harmonic Cosine Note the average is zero (anti-symmetric about centre)
41Mixer output for the 3rd Harmonic Cosine Note the average is zero (anti-symmetric about centre)
42Summary of the components of a Square wave We have seen that you do get the 1/3, 1/5 1/7 ratios (odd harmonics) we used in the signals talk.The even Sine harmonics have equal numbers of plus and minus (half wave) peaks so are zeroOdd Sine harmonics all have two more positive peaks than negative out of a total of double their harmonic number. Hence the 1/3, 1/5, 1/7 etc. ratios.Cosine harmonics are all anti-symmetric and thus zero
43The Inverse Fourier transform In the signals talk we took the harmonics of a square wave and combined them.This was an Inverse Fourier transform!(If done correctly these transforms are reversible and lossless)We may look at the Fast Fourier transform (FFT) in a later talk. Its just a quicker way of doing Fourier transforms.