14: FM Radio Receiver

Transcription

1 (1) (2) (3) DSP and Digital Filters ( ) FM Radio: 14 1 / 12

2 (1) (2) (3) FM spectrum: 87.5 to 108 MHz Each channel: ±100 khz Baseband signal: Mono (L + R): ±15kHz Pilot tone: 19 khz Stereo (L R): 38±15kHz RDS: 57±2kHz FM Modulation: Freq deviation: ±75 khz L R signal is multiplied by 38kHz to shift it to baseband [This example is taken from Ch 13 of Harris: Multirate Signal Processing] DSP and Digital Filters ( ) FM Radio: 14 2 / 12

3 (1) (2) (3) FM band: 87.5 to 108 MHz Normally sample at f s > 2f However: f s = 80MHz aliases band down to [7.5, 28]MHz. ve frequencies alias to [ 28, 7.5] MHz. We must suppress other frequencies that alias to the range ±[7.5, 28] MHz. Need an analogue bandpass filter to extract the FM band. Transition band mid-points are at f s = 80MHz and 1.5f s = 120MHz. You can use an aliased analog-digital converter (ADC) provided that the target band fits entirely between two consecutive multiples of 1 2 f s. Lower ADC sample rate. Image = undistorted frequency-shifted copy. DSP and Digital Filters ( ) FM Radio: 14 3 / 12

4 (1) (2) (3) FM band shifted to 7.5 to 28MHz (from 87.5 to 108MHz) We need to select a single channel 200kHz wide We shift selected channel to DC and then downsample to f s = 400kHz. Assume channel centre frequency is f c = c 100kHz We must apply a filter before downsampling to remove unwanted images The downsampled signal is complex since positive and negative frequencies contain different information. We will look at three methods: 1 Freq shift, then polyphase lowpass filter 2 Polyphase bandpass complex filter 3 Polyphase bandpass real filter DSP and Digital Filters ( ) FM Radio: 14 4 / 12

5 (1) (1) (2) (3) Multiply by e j2πr fc 80MHz to shift channel at f c to DC. f c = c 100k f c 80M = c 800 Result of multiplication is complex (thick lines on diagram) Next, lowpass filter to ±100kHz ω = 2π 200 k 80 M = M = 60 db 3.5 ω = 1091 Finally, downsample 200 : 1 Polyphase: H p (z) has = 6 taps Complex data Real Coefficients (needs 2 multiplies per tap) Multiplication Load: 2 80MHz (freq shift) MHz (H p (z)) = 14 80MHz DSP and Digital Filters ( ) FM Radio: 14 5 / 12

6 (2) (1) (2) (3) Channel centre frequency f c = c 100kHz where c is an integer. Write c = 4k +l where k = c 4 and l = cmod 4 We multiply u[r] by e j2πr c 800, convolve with h[m] and then downsample: v[n] = M m=0 h[m]u[200n m]e j2π(200n m) c 800 [r = 200n] = M mc 4k+l j2π200n m=0h[m]ej2π 800u[200n m]e 800 [c = 4k +1] = M m=0 g [c][m]u[200n m]e j2π ln 4 [g [c] [m] = mc j2π h[m]e 800 ] = ( j) ln M m=0 g [c][m]u[200n m] [e j2π ln 4 indep of m] Multiplication Load for polyphase implementation: G [c],p (z) has complex coefficients real input 2 mults per tap ( j) ln {+1, j, 1, +j} so no actual multiplies needed Total: 12 80MHz (for G [c],p (z)) + 0 (for j ln ) = 12 80MHz DSP and Digital Filters ( ) FM Radio: 14 6 / 12

7 (3) (1) (2) (3) Channel frequency f c = c 100kHz where c = 4k +l is an integer cm j2π g [c] [m] = h[m]e 800 c(200s+p) j2π g [c],p [s] = g c [200s+p]= h[200s+p]e 800 [polyphase] = h[200s+p]e j2π cs cp 4 j2π e 800 h[200s+p]e j2π cs 4 α p (4k+l)s j2π Define f [c],p [s] = h[200s+p]e 4 = j ls h[200s+p] Although f [c],p [s] is complex it requires only one multiplication per tap because each tap is either purely real or purely imaginary. Multiplication Load: cp j2π 6 80 MHz (F p (z)) MHz ( e 800) = 10 80MHz DSP and Digital Filters ( ) FM Radio: 14 7 / 12

8 (1) (2) (3) Complex FM signal centred at DC: v(t) = v(t) e jφ(t) We know that logv = log v +jφ The instantaneous frequency of v(t) is dφ dt. We need to calculate x(t) = dφ dt = di(logv) dt = I ( 1 v ) dv dt = 1 v I ( ) v dv 2 dt We need: (1) Differentiation filter, D(z) (2) Complex multiply, w[n] v [n] (only need I part) (3) Real Divide by v 2 x[n] is baseband signal (real): DSP and Digital Filters ( ) FM Radio: 14 8 / 12

9 (1) (2) (3) Differentiation Filter Window design method: (1) calculate d[n] for the ideal filter (2) multiply by a window to give finite support Differentiation: d dt ejωt = jωe jωt D(e jω ) = Hence d[n] = 1 2π ω0 ω 0 jωe jωn dω = j 2π [ ωe jnω jn ejnω j 2 n 2 ] ω0 = nω 0 cosnω 0 sinnω 0 πn 2 { jω ω ω 0 0 ω > ω 0 ω 0 [IDTFT] 1.5 ω 0 0 H H (db) ω ω (rad/sample) ω (rad/sample) Using M = 18, Kaiser window, β = 7 and ω 0 = 2.2 = 2π 140kHz 400 khz : Near perfect differentiation for ω 1.6 ( 100kHz for f s = 400kHz) Broad transition region allows shorter filter DSP and Digital Filters ( ) FM Radio: 14 9 / 12

10 (1) (2) (3) Pilot tone extraction Aim: extract 19kHz pilot tone, double freq real 38kHz tone. 20kHz j2πn (1) shift spectrum down by 20kHz: multiply by e 400 khz (2) low pass filter to ±1kHz to extract complex pilot at 1kHz: H(z) (3) square to double frequency to 2kHz [ ( e jωt) 2 = e j2ωt ] 40kHz +j2πn (4) shift spectrum up by 40kHz: multiply by e 400 khz (5) take real part More efficient to do low pass filtering at a low sample rate: Transition bands: F(z): 1 19kHz, H(z): 1 3kHz, G(z): 2 18kHz ω = 0.28 M = 60, ω = , ω = DSP and Digital Filters ( ) FM Radio: / 12

11 (1) (2) (3) Polyphase Pilot tone Anti-alias filter: F(z) Each branch, F p (z), gets every 20 th sample and an identical e j2π n 20 So F p (z) can filter a real signal and then multiply by fixed e j2π p 20 Anti-image filter: G(z) Each branch, G p (z), multiplied by identical e j2π n 10 So G p (z) can filter a real signal Multiplies: F and G each: (3+2) 400kHz, H +x 2 : (2 30+4) 20kHz Total: khz [Full-rate H(z) needs khz] DSP and Digital Filters ( ) FM Radio: / 12

12 (1) (2) (3) allows sampling below the Nyquist frequency Only works because the wanted signal fits entirely within a Nyquist band image Polyphase filter can be combined with complex multiplications to select the desired image subsequent multiplication by j ln shifts by the desired multiple of 1 4 sample rate No actual multiplications required FM demodulation uses a differentiation filter to calculate dφ dt Pilot tone bandpass filter has narrow bandwidth so better done at a low sample rate double the frequency of a complex tone by squaring it This example is taken from Harris: 13. DSP and Digital Filters ( ) FM Radio: / 12