SMOOTHING AND WINDOWING 71 for the improved attenuation of the side lobes. For the Hann the value of the peak response is −38 dB at k ≈2.4, for the Hammimg it is −53 dB at k ≈2.2, and for the rectangular it is only −18 dB at k ≈2.5. Comparing the main lobe widths in the vicinity of X (k) =1, the Hann is 1.46 at −20 dB and the Hamming is 1.36 at −20 dB, which can perhaps be a worthwhile improvement. Comparing the Hamming and the Hann, the Hamming provides deeper attenuation of the Þrst side-lobe (which is one of its main goals) and limits in the neighborhood of 45 to 60 dB at the higher-frequency lobe peaks (another goal). For many applications this is quite satisfactory. On the other hand, the Hann is not quite as good up close but is much better at higher frequencies, and this is often preferred. In many introductory references, these two windows seem to meet the majority of practical requirements for non-integer frequency (k) values. In the equations for the Hamming and Hann windows we see the sum of a constant term and a cosine term. There are other window types, such as the Kaiser window and its variations, that have additional cosine terms. These may be found in the references at the end of the chapter and are not pursued further in this book. These other window types are useful in certain applications, as discussed in the references. We noted that the time-domain window sequence multiplies a time- domain signal sequence. In the frequency domain the spectrum of the window convolves with the spectrum of the signal. These interesting sub- jects will be explored in Chapter 5. Figure 4-4 is a modiÞcation of Fig. 4-3 that illustrates the use of con- volution in the frequency domain. Equation (4-3) contains formulas for the spectra of the windows, including frequency translation to 38.0. Rectangle: X 1 (k) = 1 N N−1  n=0  (1)  exp  j2π n N (38.0 −k)  Hamming: X 2 (k) = 1 N N−1  n=0  0.54 − 0.56  cos  2π n N − 1  ×  exp  j2π n N (38.0 −k)  (4-3) 72 DISCRETE-SIGNAL ANALYSIS AND DESIGN Hann: X 3 (k) = 1 N N−1  n=0  1 2  1 −  cos  2π n N − 1  ×  exp  j2π n N (38.0 −k)  The two-sided baseband signal is translated up to a center frequency of +38.0, where it shows up as a lower sideband and an upper sideband. The way that the Hamming dominates from 38.0 to 41.0 and the Hanning takes over starting at 42 is quite noticeable. The performance of the Hamming 30 32 34 36 38 40 42 44 4 6 −60 −50 −40 −30 −20 −10 0 k dB Rectangular Hamming Hanning (a) 38 39 40 41 42 43 −60 −50 −40 −30 −20 −10 0 k dB Rectangular Hamming Hanning (b) Figure 4-4 (a) Rectangular, Hanning, and Hamming windows translated to k = 38.0. (b) Close-up of part (a) showing window behavior between integer k values. SMOOTHING AND WINDOWING 73 from 40 to 41 is also interesting. Chapter 8 shows how a single-sideband spectrum (USSB or LSSB) for these windows can be generated. Referring again to Fig. 4-3, it is apparent that if we can stay away from k < 2, the Hann and Hamming are more tolerant of frequency departures from integer values. This should be considered when designing an exper- iment or when processing experimental data or a communication signal. If the number of n and k values can be doubled, the resolution can be improved so that after adjusting the frequency scaling factor, k =2 repre- sents a smaller actual frequency difference. If a certain positive frequency range 0 to 10 kHz is needed and an N value of 256, or 128 positive fre- quencies, is chosen, the resolution is 78 Hz per bin, and k =2 corresponds to 156 Hz, which may not be good enough. For N =1024 (512 positive), the resolution is 19.5 Hz and k =2 corresponds to 39 Hz, which is a lot better. Increasing N would also seem to make the close alignment with integer values more desirable. But in the Hamming example the sidelobe peaks are better than 43 dB below the k =0 level, which is often good enough, and means that alignment with integer (k) values may be completely unnecessary (compare this with the rectangular window). This reduction of lobe peaks and the reduced need for integer (k ) values is the major goal of window “carpentry.” Note also that the k =0 value is about 5 dB below the 0-dB reference level, and a gain factor of 5 dB can be included in the design to compensate. The operations just concluded can be extended to multiple input signals. Equation (4-2) can be restated as follows: y(n) = w(n)(x 1 (n) + x 2 (n) +···) = w(n)x 1 (n) +w(n)x 2 (n) +··· Y(k) = W(k)∗ (X 1 (k) +X 2 (k) ···) = W(k) ∗ X 1 (k) +W(k) ∗ X 2 (k) ··· (4-4) where ∗ is the convolution operator. This means that multiplication of a window time sequence and the sum of several signal time sequences is a distributive operation, and the convolution of their spectra is also a distributive operation. Any window function performs the same operation 74 DISCRETE-SIGNAL ANALYSIS AND DESIGN 33 34 35 36 37 38 39 40 41 42 43 44 45 −50 −40 −30 −20 −10 0 10 k (a) (b) dB Rectangular Hamming Hanning 33 34 35 36 37 38 39 40 41 42 43 44 45 −50 −40 −30 −20 −10 0 10 k dB Rectangular Hamming Hanning Figure 4-5 Two-tone input signal: (a) with 2 units of frequency separa- tion; (b) with 3 units of frequency separation. on each of the signal functions in the time domain and also in the fre- quency domain. This is mentioned because it may not be immediately obvious. An illustration of this is shown in Fig. 4-5a for two signals at 38.0 and 40.0 (poor separation) and in Fig. 4-5b for 37.5 and 40.5 (better separation), for each of the three window functions. Using the frequency scaling factors as described previously, the resolution can be adjusted as required. At certain other non-integer close separations it will be noticed that adjacent lobe peaks interact and deform each other slightly (the reader is encouraged to try this). SMOOTHING AND WINDOWING 75 The frequency conversions in Figs. 4-4 and 4-5 are exactly identi- cal to the idealized “mixer circuit” found in radio textbooks, where a positive-frequency baseband signal and a positive-frequency local oscillator (L.O.) are multiplied together to produce upper and lower side- bands about the L.O. frequency with a suppressed L.O. frequency content. The frequency conversion itself is a second-order nonlinear process, as Eq. (4-3) conÞrms, but the two mixer sideband output amplitudes are each linearly related to the baseband input amplitude if the L.O. level is assumed to be constant, hence the colloquial term “linear mixer.” Actual mixer circuits are not exactly linear in this manner. We also men- tioned previously the two-sided baseband spectrum being translated to produce a double-sideband output. Both concepts do the same thing in the same way. There are situations where the signal data extend over a long time period. The analysis can be performed over a set of smaller windowed time periods that intersect coherently so that the overall analysis is cor- rect. The article by Harris [1978] is especially excellent for this topic and for the subject of windows in general; see also [Oppenheim and Schafer, 1975]. A frequent problem involves sudden transitions in the amplitude values between the end (or beginning) of one time sequence and the beginning (or end) of the next. This causes a degradation of the spectrum due to the introduction of excessive undesired components and also signiÞcant aliasing problems. The methods described in the smoothing section of the chapter can take care of this problem using the following guidelines: (1) create a nearly-zero amplitude guardband at each end of the sequence; (2) perform one or more three-point smoothing operations on the time and/or frequency data; (3) use scaling techniques to get the required time and frequency coverage and resolution; and (4) use a window of the type discussed in this segment to reduce the need for exact integer values of time and frequency. We see also that the Hamming and Hanning time-domain windows are zero or almost zero at the edges, which improves protection against aliasing in the time domain. Frequency-domain aliasing is also improved, especially with the Hanning window, as the spectrum plots show. Other window types, such as the Kaiser, can be compared for these properties. . 1  ×  exp  j2π n N (38.0 −k)  (4-3) 72 DISCRETE-SIGNAL ANALYSIS AND DESIGN Hann: X 3 (k) = 1 N N−1  n=0  1 2  1 −  cos  2π n N − 1  ×  exp  j2π n N (38.0 −k)  The two-sided baseband signal is translated. center frequency of +38.0, where it shows up as a lower sideband and an upper sideband. The way that the Hamming dominates from 38.0 to 41.0 and the Hanning takes over starting at 42 is quite noticeable textbooks, where a positive-frequency baseband signal and a positive-frequency local oscillator (L.O.) are multiplied together to produce upper and lower side- bands about the L.O. frequency with a